Computation of All the Amicable Pairs Below 1010 · Computation of All the Amicable Pairs Below 1010 By H. J. J. te Riele Abstract. An efficient exhaustive numerical search method

MATHEMATICS OF COMPUTATIONVOLUME 47. NUMBER 175JULY I4K6. PACIES .161-36«

Computation of All the Amicable Pairs Below 1010

By H. J. J. te Riele

Abstract. An efficient exhaustive numerical search method for amicable pairs is.described.

With the aid of this method all 1427 amicable pairs with smaller member below 1010 have

been computed, more than 800 pairs being new. This extends previous exhaustive work below

108 by H. Cohen. In three appendices (contained in the supplements section of this issue),

various statistics are given, including an ordered list of all the gcd's of the 1427 amicable pairs

below 1010 (which may be useful in further amicable pair research). Suggested by the

numerical results, a theorem of Borho and Hoffmann for constructing APs has been extended.

1. Introduction. Let a(m) denote the sum of all the divisors of m, including 1 and

m. An amicable pair (AP) is a pair of positive integers (m,n), m < n, such that

a(m) = a(n) = m + n. We note that m is abundant (since a(m) > 2m) and that n

is deficient (since a(n) < 2n). The smallest AP is

(220,284) = (225.11,2271).In order to check whether or not a given positive integer m is the smaller member

of an amicable pair, it seems necessary, at first sight, to compute a(m) and

n := a(m) — m, to check whether n > m (i.e., whether m is abundant), and, if so, to

compute a(n) and compare a(m) with a(n). This involves one or two complete

factorizations, in case m is deficient or abundant, respectively. However, a closer

look reveals that it is often possible to find out whether a given number m is

deficient (hence cannot be the smaller member of an AP) without the need to

factorize it completely. Moreover, once o(m) and n (= a(m) - m) have been

computed, it is often possible to discover that a(n) ¥= a(m) without the need to

factorize n completely.

These considerations have guided the design of an efficient exhaustive numerical

AP search algorithm, the details of which are given in Section 2. With the aid of this

algorithm we have extended Cohen's exhaustive list of all 236 APs with smaller

member below 108 [4] to all 1427 APs with smaller member below 1010. Of these,

601 have been published earlier [6], [7]. The other 826 seem to be new, and are

published here for the first time (9 of them have been communicated to the author

already in 1983 and 1984 by Woods (2), Borho (2) and Lee (5)). Section 3 presents

details of the computations together with several tables collected from this search.

Moreover, a result of Borho and Hoffmann for constructing APs is extended, as was

suggested by the numerical tables.

Received March 27, 1985.

1980 Mathematics Subject Classification. Primary 10A40; Secondary 10-04.

Key words and phrases. Amicable pair.

©1986 American Mathematical Society

0025-5718/86 $1.00 + $.25 per page

361

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362 H. J. J TE RIELE

Three appendices to this paper appear in the supplements section of this issue.

These may also be obtained by writing to the author.

In Appendix I, we present the complete list of all 1427 APs with smaller member

below 1010 ordered according to the size of the smaller members of the pairs.

Appendix II displays the same list with a different ordering, viz., according to the

various occurring types (defined in Section 3). Finally, Appendix III tabulates all the

greatest common divisors of the 1427 APs, in increasing order, together with their

frequencies of occurrence, and, for each gcd g, the rank numbers of all the APs

(m, n) for which gcd(w, n) = g.

2. Check Whether a Given m is the Smaller Member of an AP. Let p, be the ;th

prime, P,/= Wk%T1 pk, Qu-= UfJ/1 pk/(pk - 1). We start with the following

lemma which gives an upper bound for a(m)/m.

Lemma 2.1. // m only has prime divisors > p¡ (i > 1) and if m < P¡¡+i (j > 1)

then a(m)/m < Q(j.

Proof. Since m < P¡¡+i = PiP¡+\ ' • • pi+j, and since any prime divisor of m is

> p¡, it follows that m has at most j different prime divisors > p¡ (otherwise we

would have m > p¡p¡+\ ••• pi+j = Pij+i)- This implies that

a(m) n pe+l ~~ i n P ~ P e o P ' rr ^k - n

~™~ = /i p'(p - i) ~ p'I ^P31" < M P7^ * k=\ Jk~^=Q,J' DIn the algorithm below, this lemma is invoked very frequently. Therefore, we require

a precomputed table of P- and g-values, large enough so that the values needed can

be found quickly by simple table look-ups.

Now we describe an efficient algorithm to check whether a given positive integer

m belongs to an AP (m, n) with m < n. This algorithm is based on the observation

that when, for given y and N, we want to verify one of the relations a(N)/N > y,

= y, < y, and when the primes 2,3,...,p have been tried as divisors of N, it may

be possible

(i) to detect, with Lemma 2.1, whether a(N)/N < y by using the information that

the unfactored portion of N only has prime divisors > p, and

(ii) to detect whether a(N)/N > y by using the factored portion of N.

In this way, much unnecessary factorization time may be avoided. The price to pay

for this gain lies in the time needed to consult the P- and Q-tables used in Lemma

2.1. In the algorithm, the index z'max is the maximum value of i for which Lemma 2.1

is invoked. In order to restrict this table look-up time, t'max should not be chosen too

large. The optimal value of z'max also depends on the actual implementation of the

algorithm (cf. Section 3).

Algorithm to Check Whether m is the Smaller Member of an AP.

Step 1. (Find out whether m is abundant; in this step, keep m = mxm2 where

gcd(m1,wi2)= 1, mx is the factored and m2 is the unfactored portion of m,

a := a(mx)/mx; start with mx := 1, m2:= m, a.= 1.)

Start factoring m by trial dividing m2 by the primes px, p2,... < m1/2. In case a

prime power divisor p'_x (e > 1) of m2 has been found, update mx, m2 and a

(mx:= mxpf_x,m2:= m/mx,a:= a • o(pf_x)/pf_x). After the trial division with

p _, (whether or not p¡_x divides m2): if a < 2 and 4 < / < ;'max, check whether m


AMICABLE PAIRS BELOW 1010 363

is possibly deficient as follows: by inspecting the P-table find the smallest value of j

(='j*) such that m2 < Pjt+i', if «ô,,7» < 2, then STOP (because, in that case, m is

deficient: by Lemma 2.1 we have a(m2)/m2 < Qijt so that

o(m) °(mx) a(m2) a(m2)-=-•-= a-< aQH» < 2).

m mx m2 m2 J

If a > 2, or / < 4 or i > i'max, the deficiency check on m is left out. After the

complete factorization of m (and simultaneous computation of o(m))\ if m < a (m)

— m =.n (i.e., m is abundant), go to Step 2, otherwise STOP.

End of Step 1

Step 2. (Given m, a(m) and n = a(m) - m, check whether a(n) = a(m); during

the factorization of n try to exclude those m for which a(n)¥= a(m) as early as

possible by testing whether a(n)/n *= ß where ß = a(m)/n; in this step, keep

n — nxn2, where gcd(«l5 n2) = 1, nx is the factored and n2 the unfactored portion

of n, a.= a(nx)/nx; start with nx:= 1, n2:= n, a.= 1.)

Start factoring n by trial dividing n2 by the primes px, p2,... < n1/2. In case a

prime power divisor pf_x (e > 1) of n2 has been found, update nx, n2 and a: if the

updated a satisfies a > ß, then STOP (because, in that case, we have

o(n) o(nx) a(n2) a(nx) a(m)-=-^-= a > ß =-,

n nx n2 nx n

so that a(n) =£ a(m)). After the trial division with /?,_, (whether or not p¡_x divides

n2): if 4 < i < ¿max check whether a(n)/n < ß as follows: by inspecting the P-table

find the smallest value of ; (=:/*) such that n2 < Pij+1. If aQijt < ß, then STOP

(because, in that case, a(n)/n < ß: by Lemma 2.1 we have a(n2)/n2 < Q¡ ,, so

that

o(n) o(nx) a(n2) a(n2)——- =-= a-< aQ, ,» < ß ).

n nx n2 n2 *'•' HJ

If / < 4 or i > ;max, the check on a(n)/n < ß is omitted. After the complete

factorization of n (and simultaneous computation of o(n))\ check whether a(n) =

a(m). If so, (m, n) is an AP.

End of Step 2

3. Computing All the APs Below 1010. In order to compute all the APs (m, n) with

m < n and 108 < m < 1010 (thus extending H. Cohen's computations reported in

[4]), we distinguish between m = 0 (mod 6) (the easy case), and m # 0 (mod 6) (the

hard case).

If m = 0 (mod6) and n = a(m) - m is even, then (m, n) cannot be an AP [5].

Therefore, n should be odd. In that case, we have [6] m = 2)íM2, n = N2, with

H e N, M and N being odd. For all the numbers m = 2'"Ai2 with 3|M and

108 < m < 1010, we computed n := a(m) - m and checked whether n was a perfect

square. Not a single such case was found. Computer time was about 6 CPU seconds.

For all m # 0 (mod 6) with 108 < m < 1010 we used the algorithm of Section 2 to

find all APs in this range. The optimal choice of jmax for our FORTRAN-implemen-

tation on a CYBER 750 was about 75. This value was chosen to be fixed for the

whole range. The speed-up factor of our program was about 15, compared with a


364 H. J. J. TE RIELE

straightforward program which, given m, computes a(m) and, if n:= a(m) - m >

m, computes a(n). A slight increase of the speed was obtained as follows. In Step 1,

in case a prime (power) factor of m2 was found and mx and a(mx) (among others)

were updated, it was checked whether both mx and o(mx) were divisible by one of

the primitive abundant numbers 20 = 225, 28 = 227, 70 = 2.5.7 and 88 = 2311. If

so, the algorithm was stopped since this implied that also m and o(m), hence also

n = a(m) — m were divisible by this abundant number, so that both m and n were

abundant. This is impossible for an AP (m, n).

The total time to cover the range 108 < m < 1010 was about 1000 (low priority)

CPU hours, spent in the last seven months of 1984.

The total number of APs (m,n) found with m < n and 108 < m < 1010 was

1191. In Appendix I (of the supplements section) all the APs with smaller member

< 1010 are given (including the 236 APs with smaller member < 108). For each

pair we list the decimal representation and the prime factorization of the members, a

rank number, a code (letter plus digit) referring to the discoverer, and the type of the

pair (defined below). For example, pair #1427 reads as follows:

1427 9967523980 2E2.257.5.17.37.3083

R942 12890541236 2E2.257.107.117191.

Table 1 gives the meaning of the codes, and their frequencies of occurrence.

Extensive information about the sources of the pairs with code LI is given in the

survey paper [6].

There are 1015 pairs with even members and 412 with odd members. The minimal

and maximal values of m/n are 0.6979 and 0.999858 for the APs #567 and #1010,

respectively.

Let A(x) be the number of APs (m, n) with m < n and m < x. From the list of

APs with m < 108, Bratley et al. [3] concluded that for x < 108, A(x) is approxi-

mately proportional to x1/2/ln(x). In Table 2 we give, for x = k. 109 (1 < k < 10):

A(x), A(x)ln(x)/xl/2, A(x)(\n(x))2/x1/2 and A(x)(\n(x))3/xl/2. From these fig-

ures we may draw the conclusion that for x < 1010, A(x) is approximately propor-

tional to x1/2/(\n(x))3.

Table 1

Status list of the first 1427 APs (m,n), m < n, with m < 1010

code #APs references and remarks

LI

R2Wl

R3

W2R6

W3

L2B4

R9

5081

73

19

11

1

52

816

[6]

[9] (#1056)sent to the author by D. Woods on June 29,1982 and published in

[7]found by the author with the methods described in [8], and

published in [7]

sent in by D. Woods on Feb. 16,1983 (#330)

found by the author in May, 1983 (#1375)

sent in by D. Woods on My 11,1983 (#1050)

sent in by E. J. Lee in July, 1984 (# #778,860,894,1241,1261)sent in by W. Borho on Nov. 2,1984 (# #809,1393)

found by the author during the systematic search described in this

paper


AMICABLE PAIRS BELOW 1010 365

Table 2

Comparison of A(x) with x1/2/(\n(x))', i = 1,2,3

A'/IO" A(x) A(x)\n(x)/x^ A(,x)(ln(x))2/x1/2 A(x)<\vt(x))i/x 1/2

1

2

34

56

7

S

9

10

586762

8981009

1100118512561317

13771427

0.38400.3649

0.35780.3527

0.34740.3444

0.34030.33580.33270.3286

7.9587.8157.8077.799

7.7597.7557.7157.6567.6257.566

164.9167.4

170.4172.4

173.3174.6

174.9174.6

174.8174.2

We define an AP (m, n), m < n, to be a regular amicable pair of type (i, j), if

(m, n) = (gM, gN), where g = gcd(w, n), gcd(g, M) = gcd(g, N) = 1, M and N

are squarefree, and the numbers of prime factors of M and TV are i and j,

respectively. Other pairs are called irregular or exotic. There are 1082 regular and

345 irregular APs with smaller member < 1010. It is easy to see that there are no

regular pairs of type (1, j), j > 1: let g be the gcd of such an AP, so that

(m,n) = (gp,gN) where p is a prime and gcd(g, p) = gcd(g,N) = 1. We have

m < n, hence p < N. By definition, a(gp) = o(gN), implying that p + 1 = a(N).

Since, for any N e M, a(N)> N, this implies that p + 1 > N, a contradiction. We

note that in this argument N need not be squarefree.

In Table 3 we give the frequency distribution of the various types among the first

1082 regular APs. We note that there are relatively few regular APs of type (j, 1),

/' > 2, and of type (/', j) with /' < j.

In [7] the total number of known APs with smaller member < 1010 was 601

(these are the APs belonging to the first four codes in Table 1). Among them were

104 irregular APs, i.e., 17.3%. Comparing this figure with the 345 irregular APs in

our complete list of APs with smaller member < 1010, i.e., 24.2%, we see that

relatively many irregular APs were found in our systematic search.

In Appendix II (of the supplements section) we present lists of all the 1082 regular

APs arranged according to their types, together with a list of the 345 exotic APs.

This appendix may be useful for searches of APs of a special type.

The regular pairs of type (;', 1), /' > 2, play an important role as "mother" pairs in

methods to generate new APs from given pairs. In [8] a substantial part of the new

APs found there was constructed from such mother pairs. In [1], Borho and

Hoffmann have partially generalized the methods from [8] by introducing the

concept of a breeder: a breeder is a pair of positive integers (ax, a2) such that the

equations

ax + a2x = a(ax) = a(a2)(x + 1)



Table 3

Frequency distribution of the first 1082 regular APs

of type (i,j), i > 2, j> 1

column

totals

3

20

16

1

0

67

271

78

6

21

280201

IX

4

24

63

7

37 422 520 98

row totals

112

591

345

34

1082

have a positive integer solution x. If x is a prime, then (ax, a2x) is an amicable pair.

For certain breeders, called "special" breeders, Borho and Hoffmann formulate the

following

Theorem 1 [1]. Let (ax,a2) be a special breeder, i.e., ax = au, a2 = a, with

gcd(a, u) = 1. Take any factorization of C:= a(u)(u + o(u) - 1) into two different

factors Dx, D2 (C = DXD2). Then, if the numbers s¡ = D¡ + a(u) - 1, for i = 1, 2,

and also q = u + sx + s2 are primes not dividing a, then (auq, asxs2) is an amicable

pair. D

Regular APs of type (/, 1), i > 2, are of the form (au,ap), p prime, and the

numbers (au, a) are special breeders which generally produce many APs with the

above theorem.

In our list of 1427 APs we found a few APs, e.g., #647 and #955, which

suggested that the condition gcà(a, u) = 1 in Theorem 1 may be dropped. In fact,

we have

Theorem 2. Let (au, a) be a breeder, i.e., there exists a positive integer x such that

au + ax = o(au) = a(a)(x + 1). Take any factorization ofC:= (x + l)(x + u) into

two different factors Dx, D2(C = DXD2). Then, if the numbers 5, = D¡ + x, for i = 1,

2, and also q = u + sx + s2 are primes not dividing a, then (auq, asxs2) is an

amicable pair. D

The proof of this theorem is left to the reader.

If gcd(a, u) = 1, then a(au) = a(a)a(u), so that x = a(u) - 1 and Theorem 2

reduces to Theorem 1. As an example, AP #955 gives the breeder (au, a) with

a = 3.5.7.19 and u = 7.29.47.181. Theorem 2 yields 16 new APs with this breeder as

input.

It is known [5] that most even APs have a pair sum which is = 0 (mod 9). Our

search proves that indeed Poulet's pair #503: (24331.19.6619,24331.199.661) is the

smallest exceptional pair. All known exceptional pairs had members = 7 (mod 9)

and a pair sum = 5 (mod 9). In our search, we found two even APs with pair sum

= 3 (mod 9), viz., the (irregular) pairs:

ÄV77.94/l92103.1627 and #874.22i9/l3237.43.139*577' 2 \3847.16763 ^ #™' " 41.151.6709.


AMICABLE pairs BELOW 10 367

Table 4

The 17 APs among the first 1427', whose pair sum is ^ 0 (mod 9)

regular

irregular

even members odd members

#503, type (2,2)#1031, type (2,2)

#1081, type (2,2)

#899, type (3,2)#1057, type (2,2)

#1158, type (3,2)

##577,874 ##7,38,78,113,256,440,1083,1175,1380

Table 5

All (37) pairs from the first 1427 APs having the same pair sum

rank numbers pair sum

prime decomposition of the pair sum,

i.e., exponents belonging to the primes

2 3 5 7 11 13 17 19 23 29 31 37

32 35105 109

137 138172 173272 276

282 286350 351347 355373 375

368 377395 399411 415

427 433462 476486 491

574 582626 630

653 665695 697717 730751 753

798 807786 787

824 840940 941926 952

997 998

1012 1019

1069 10971124 1142

1147 1150

1143 11811232 12331254 1265

1249 12551272 1278

1410 1425

129600020528640

3773952075479040

321408000

348364800556839360579156480638668800

661893120761177088796340160

8838720001181174400

1282417920

20684160002395008000

2682408960

315502387235997696004049740800

4606156800

4716601344

509483520068245632006897623040

79252992008273664000

1002792960011195712000

114162048001209821184013473008640

14341017600

14478912000

1505806848019926466560

7

9

10

31

1 1

11 4 1 110 4

13 56 6

912

12

10

6

8

7

8

9

10

12

11

13

10

13

13

109

13

11

11

3

2 11 1

5 1 24 2 1

5 1 1

1

1 2

3

2 21 13 1

27

37

5

512 59 39

12

10

12

9

10

6 2 1

3 2 2

1 1

2 2

3 1232 1

11

2

3

1

14 5 1 1 1



These are the first two examples of APs of the form described in [5, Theorem I, case

(b)] (also cf. the remarks immediately following Table I in [5]). Table 4 gives the

rank numbers of the 17 APs with smaller member < 1010 whose pair sum is # 0

(mod 9), divided into even and odd pairs, and regular and irregular pairs.

Another question, suggested by Professor C. Pomerance, is whether pairs, triples,

quadruples, etc. of APs exist having the same pair sum. Among the first 1427 APs,

we found 37 such pairs of APs, but no such triples, quadruples, etc. Table 5 gives the

rank numbers of these pairs of APs, and the prime factorization of their pair sums.

The pair sums only have prime divisors < 37. In 30 of the 37 cases at least one

member of the pair was found during the exhaustive search described in the present

paper.

In Appendix III (of the supplements section) we tabulate all the greatest common

divisors of the first 1427 APs, ordered according to their size, with frequencies, and

with the rank numbers of all the APs corresponding to a given gcd. This might be

useful in further searches for special APs, and in searches for so-called isotopic APs

(cf., [6, p. 83]). For example, new APs, isotopic with APs from the list of 1427 APs,

are obtained by replacing the common factor 335 in # #882 and 1087 by 327.13, by

replacing the common factor 3353 in #1205 by 325231, and by replacing the

common factor 335231 in ##717 and 1228 by 365.23.137.547.1093, and by

3105.23.107.3851.

In [8], we have presented methods to find new APs from known APs. By applying

these methods to the new APs among the first 1427 APs, we have found 117 new

APs (with smaller member > 1010). The new APs were found mainly from mother

pairs having a relatively simple structure, like those of type (/', 1), i > 1. They will be

published in a forthcoming report [2], together with many other new amicable pairs.

Centre for Mathematics and Computer Science

Kruislaan 413

1098 SJ Amsterdam

The Netherlands

1. W. Borho & H. Hoffmann, "Breeding amicable numbers in abundance," Math. Comp., v. 46, 1986,

pp. 281-293.2. W. Borho, H. Hoffmann & H. J. J. te Riele, Table of Amicable Pairs Between 1010 and 1052,

CWI-report. (In preparation.)

3. P. Bratley, F. Lunnon & John McKay, "Amicable numbers and their distribution," Math. Comp.,

v. 24,1970, pp. 431-432.4. H. Cohen, "On amicable and sociable numbers," Math. Comp., v. 24, 1970, pp. 423-429.

5. E. J. Lee, " On divisibility by nine of the sums of even amicable pairs," Math. Comp., v. 23,1969, pp.

545-548.6. E. J. Lee & J. S. Madachy, "The history and discovery of amicable numbers," J. Recreational

Math., v. 5, 1972; Part I: pp. 77-93, Part II: pp. 153-173, Part III: pp. 231-249.

7. H. J. J. TE RlELE, Table of 1869 New Amicable Pairs Generated from 1575 Mother Pairs, Report NN

27/82, Math. Centre, Amsterdam, Oct. 1982.

8. H. J. J. te Riele, "On generating new amicable pairs from given amicable pairs," Math. Comp., v.

42, 1984, pp. 219-223.9. H. J. J. TE RlELE, Further Results on Unitary Aliquot Sequences, Report NW 2/78, Math. Centre,

Amsterdam, 2nd ed., Jan. 1978.


MATHEMATICS OF COMPUTATIONVOLUME 47. NUMBER 175JULY \1Xb. I'A(iESS9-S40

Supplement to

Computation of All the Amicable Pairs Below 1010

By H. J. J. te Riele

Appendix I

The first 1427 APs

1 220 2E2.5.11 3DLI 21 284 2E2.71

2 1184 2E5.37LI X 1210 2.5.11E2

3 2620 2E2.5.131LI 22 2924 2E2.17.43

4 5020 2E2.5.251LI 22 5564 2E2.13.107

5 6232 2E3.19.41LI X 6368 2E5.199

6 10744 2E3.17.79LI 22 10856 2E3.23.59

7 12285 3E3.5.7.13LI X 14595 3.5.7.139

8 17296 2E4.23.47LI 21 18416 2E4.1151

9 63020 2E2.23.5.137LI 21 76084 2E2.23.827

10 66928 2E4.47.89LI 22 66992 2E4.53.79

11 67095 3E3.5.7.71LI 22 71145 3E3.5.17.31

12 69615 3E2.7.13.5.17LI 21 87633 3E2.7.13.107

13 79750 2.5E3.11.29LI X 88730 2.5.19.467

14 100485 3E2.5.7.11.29LI 32 124155 3E2.S.31.89

15 122265 3E2.5.13.11.19LI 21 139815 3E2.5.13.239

16 122368 2E9.239LI X 123152 2E4.43.179

17 141664 2E5.19.233LI X 153176 2E3.41.467

18 142310 2.5.7.19.107LI 32 168730 2.5.47.359

19 171856 2E4.23.467LI 22 176336 2E4.103.107

20 176272 2E4.23.479LI 22 180848 2E4.89.127

21 185368 2E3.17.29.47LL 32 203432 2E3.59.431

22 196724 2E2.11.17.263LI 22 202444 2E2.11.43.107

23 280540 2E2.5.13E2.83LI X 365084 2E2.187.853

24 308620 2E2.5.13.1187LI 32 389924 2E2.43.2267

.5E2.11.83

.71.4332E3.13.23.1492E3.199.251

25 319550LI X 430402

26 356408LI 32 399592

27 437456 2E4.19.1439LI 22 455344 2E4.149.191

28 469028 2E2.7E2.2393LI X 486178 2.7E2.11E2.41

29 503056 2E4.23.1367LI 22 514736 2E4.53.607

30 522405 3E2.5.13.19.47LI 22 525915 3E2.5.13.29.31

31 600392 2E3.13.23.251LI 32 669688 2E3.97.863

32 609928 2E3.11.29.239LI 32 686072 2E3.191.449

33 624184 2E3.11.41.173LI 32 691256 2E3.71.1217

34 635624 2E3.11.31.233LI 32 712216 2E3.127.701

35 643336 2E3.29.47.59LI 32 652664 2E3.17.4799

36 667964 2E2.11.17.19.47LI 43 783556 2E2.31.71.89

37 726104 2E3.17.19.281LI 32 796696 2E3.53.1879

38 802725 3.5E2.7.11.139LI X 863835 3.5.7.19.433

39 879712 2E5.37.743LI X 901424 2E4.53.1063

40 898216 2E3.11.59.173LI 32 980984 2E3.47.2609

41 947835 3E3.5.7.17.59LI 32 1125765 3E3.5.31.269

42 998104 2E3.17.41.179LI 32 1043096 2E3.23.5669

43 1077890 2.5.11.41.239LI 33 1099390

44 1154450LI 22 1189150

45 1156870LI 32 1292570

.5.17.29.223

.5E2.11.2099

.5E2.17.1399

.5.11.13.809.19.6803

46 1175265 3E2.7E2.13.5.41LI 21 1438983 3E2.7E2.13.251

47 1185376 2E5.17.2179LI X 1286744 2E3.41.3923

48 1280565 3E2.5.13.11.199LI 22 1340235 3E2.5.13.29.79

49 1328470 2.5.11.13.929LI X 1483850 2.5E2.59.503

50 1358595 3E2.5.19.7.227LI 22 1486845 3E2.5.19.37.47

51 1392368 2E4.17.5119LI 22 1464592 2E4.239.383

52 1466150 2.5E2.7.59.71LI X 1747930 2.5.47.3719

53 1468324 2E2.11.13.17.151LI 43 1749212 2E2.37.53.223

54 1511930 2.5.7.21599LI 23 1598470 2.5.19.47.179

55 1669910 2.5.11.17.19.47LI 42 2062570 2.5.239.863

56 1798875 3E3.5E3.13.41LI X 1870245 3E2.5.13.23.139

57 2082464 2E5.59.1193LI 22 2090656 2E5.79.827

58 2236570 2.5.7.89.359LI 33 2429030 2.5.23.59.179

59 2652728 2E3.13.23.1109LI 32 2941672 2E3.71.5179

60 2723792 2E4.37.43.107LI 32 2874064 2E4.263.683

©1986 American Mathematical Society

0025-5718/86 $1.00 + $.25 per page

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Appendix II

The first 1427 APs ordered according

to the various occurring types

AMICABLE PAIRS OF TYPE {2,1)

LI 218

LI 219

LI 2112

LI 2115

LI 2146

LI 21104

LI 21117

LI 21

162LI 21

291LI 21

297LI 21

303LI 21

460LI 21

629LI 21

640LI 21

792LI 21

888LI 21

1030LI 21

1191LI 21

1219LI 21

220 2E2.5.11284 2E2.7117296 2E4.23.4718416 2E4.115163020 2E2.23.5.13776084 2E2.23.82769615 3E2.7.13.5.1787633 3E2.7.13.107122265 3E2.5.13.11.19139815 3E2.5.13.2391175265 3E2.7E2.13.5.411438983 3E2.7E2.13.2519363584 2E7.191.3839437056 2E7.7372711498355 3E4.5.11.29.S912024045 3E4.5.11.269931536855 3E2.5.7.53.188932148585 3E2.5.7.102059175032884 2E2 . 13.17.389.509175826716 2E2.13.17.198899183408615 3E2.5.13.19.29.569190055385 3E2.5.13.19.17099196421715 3E2.5.19.37.7.887224703405 3E2.5.19.37.7103536637465 3E2.7E2.13.97.5.193646745463 3E2.7E2.13.97.11631191953763 3E2.7E2.11.13.41.4611223611389 3E2.7E2.11.13.194031225052829 3E4.7.11.29.13.5211321639011 3E4.7.11.29.73072172649216 2E8.257.330232181168896 2EB.85201912935281375 3E3 . 5E3.13.149.4492961518625 3E3.5E3.13.674994149106335 3E4.5.11E3.43.1794268776545 3E4.5.11E3.79196066248175 3E2 . 5E2 . 13 . 31 .149 . 4496120471825 3E2.5E2.13.31.674996370495978 2.7E2.19.23.11.135236950103062 2.7E2.19.23.162287

AMICABLE PAIRS OF TYPE (3,1):

86 6955216 2E4.19.137.167LI 31 7418864 2E4.463679

151 23358248 2E3.37.23.47.73LI 31 25233112 2E3.37.85247

164 32205616 2E4.17.167.709LI 31 34352624 2E4.2147039

196 52695376 2E4.17.151.1283LI 31 56208368 2E4.3513023

270 147366765 3E2.7E2.13.5.53.97LI 31 182028483 3E2.7E2.13.31751

312 205843365 3E2.7E2.13.5.43.167LI 31 254264283 3E2.7E2.13.44351

390 347263216 2E4.17.137.9319LI 31 370414064 2E4.23150879

446 492275992 2E3.131.13.23.1571R9 31 553544168 2E3.131.528191

648 1254255550 2.5E2.23.19.137.419R9 31 1333078850 2. 5E2.23.1159199

661 1309651310 2.5.11.29.571.719R9 31 1359071890 2.5.11.12355199

753 1957374968 2E3.31.17.107.4339LI 31 2092365832 2E3.31.8436959

782 2115211995 3E3.5.13.17.31.2287R9 31 2312891685 3E3.5.13.1317887

979 3693013664 2E5.41.131.21487LI 31 3812143072 2E5.119129471

1009 3986534090 2.5.929.7.11.5573LI 31 4971106870 2.5.929.535103

1228 6562770525 3E3.5E2.31.17.19.971R9 31 7322055075 3E3.5E2.31.349919

1300 7696871576 2E3.19.53.127.7523R9 31 7904894824 2E3.19.52005887

TOTAL NUMBER: 16

TOTAL NUMBER: 20

AMICABLE PAIRS OF TYPE (4,1):

779 2099442345 3.5.7.11.13.37.3779LI 41 2533809495 3.5.7.24131519

TOTAL NUMBER:

S22


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Appendix III

The gcd's of the first 1427 APs

GCD PREQ RANK NUMBER(S) OF AP'S WITH THIS GCD

1 3 4 23 24 36 53 64 83 97115 153 154 165 209 226 238 255 267 274294 313 323 345 347 348 502 505 570 586632 645 721 738 748 750 763 776 800 807811 822 842 846 852 867 877 909 920 937975 976 1060 1119 1132 1160 1251 1307 1316 1322

1350 1355 1365 1368 1374 1399 1406

37 40 42 47 59 62 65 67 69 7476 80 81 84 90 93 94 95 103 110

112 116 120 122 126 131 140 143 144 148150 159 161 163 174 180 182 184 186 192199 201 206 208 212 217 218 239 240 244258 266 279 280 293 296 298 300 305 311314 315 317 318 330 333 339 353 357 361363 374 375 376 382 387 400 405 417 432438 447 452 455 458 459 463 467 475 498513 523 525 546 552 553 558 571 574 579580 587 591 599 606 631 633 639 644 646652 658 665 668 690 702 713 716 720 722727 737 762 771 788 806 808 814 836 845848 854 855 858 863 869 871 876 881 912913 919 925 942 946 953 972 973 974 978981 982 992 999 1011 1026 1028 1043 1063 1077

1078 1079 1113 1117 1120 1126 1130 1141 1152 11531164 1170 1182 1184 1213 1218 1222 1235 1243 12501252 1253 1279 1302 1320 1329 1345 1354 1356 13611376 1378 1391 1393 1404 1407 1408 1417

13 18 43 45 49 52 54 55 58 7075 79 98 99 105 107 128 146 190 210

219 227 228 264 275 283 286 289 299 328332 343 378 420 437 441 448 476 479 507519 534 544 589 598 609 617 620 627 664670 674 676 723 759 769 826 834 835 843859 897 904 1001 1007 1023 1027 1042 1053 1062

1064 1066 1074 1093 1103 1104 1105 1108 1111 11231148 1159 1167 1209 1221 1240 1257 1266 1285 12871373 1394 1398 1419

25 61 73 77 119 127 135 177 187 198249 377 395 399 412 470 480 651 701 756790 809 825 857 873 1070 1110 1342 1344 1384

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