A063990 -id:A063990 - cp's OEIS frontend

A262622 Amicable pairs of even numbers.

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 17296, 18416, 63020, 76084, 66928, 66992, 79750, 88730, 122368, 123152, 141664, 153176, 142310, 168730, 171856, 176336, 176272, 180848, 185368, 203432, 196724, 202444, 280540, 365084, 308620, 389924, 319550, 430402, 356408, 399592, 437456, 455344

Offset: 1

Omar E. Pol, Nov 09 2015

If there are no amicable pairs whose members have distinct parity then this is also the even terms of A259180.
First differs from A063990, A259180, A259933 at a(13).
First differs from A262624 at a(16).

Amiram Eldar, Table of n, a(n) for n = 1..20000

Cf. A002025, A002046, A005843, A063990, A259180, A259933, A262623, A262624.

listap(nn) = {forstep(n=2, nn, 2, m = sigma(n)-n; if ((m > n) && (n==sigma(m)-m), print1(n, ", ", m, ", ")););} \\ Michel Marcus, Nov 14 2015

A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.

Original entry on oeis.org

134, 284, 410, 632, 1292, 1628, 4064, 9752, 12224, 22712, 66992, 72944, 403988, 556544, 2161664, 2330528, 8517632, 13228352, 14563832, 15422912, 20732792, 89472632, 134733824, 150511232, 283551872, 537903104, 731670272, 915473696, 1846850576, 2149548032, 2159587616

Offset: 1

Timothy L. Tiffin, Aug 16 2016

Any term x = a(m) in this sequence can be used with any term y in A275996 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable.
The smallest amicable pair is (220, 284) = (A275996(2), a(2)) = (A063990(1), A063990(2)), where 284 - 220 = 64 is the abundance of 220 and the deficiency of 284.
The amicable pair (66928, 66992) = (A275996(7), a(11)) = (A063990(18), A063990(19)), where 66992 - 66928 = 64 is the deficiency of 66992 and the abundance of 66928.
Contains numbers 2^(k-1)*(2^k + 63) whenever 2^k + 63 is prime. - Max Alekseyev, Aug 27 2025

			a(1) = 134, since 2*134 - sigma(134) = 268 - 204 = 64.

Max Alekseyev, Table of n, a(n) for n = 1..89

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A387352 (k=32).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
Cf. A002025, A063990.

Select[Range[10^7], 2 # - DivisorSigma[1, #] == 64 &] (* Michael De Vlieger, Jan 10 2017 *)

isok(n) = 2*n - sigma(n) == 64; \\ Michel Marcus, Dec 30 2016

a(23)-a(31) from Jinyuan Wang, Mar 02 2020

A291422 List of pairs of amicable numbers (m,n) where the sum of the pair is divisible by 10.

Original entry on oeis.org

6232, 6368, 10744, 10856, 12285, 14595, 66928, 66992, 67095, 71145, 79750, 88730, 100485, 124155, 122265, 139815, 122368, 123152, 141664, 153176, 142310, 168730, 176272, 180848, 185368, 203432, 356408, 399592, 437456, 455344, 522405, 525915, 600392, 669688, 609928, 686072

Offset: 1

Zoltan Galantai, Aug 22 2017

The sequence lists those amicable pairs (m,n) in increasing order where the sum of the amicable pair is divisible by ten.
Up to the first 5001 amicable pairs, 88.1% of the sums satisfy this condition (up to the first 100 amicable pairs: 74%; up to the first 1000: 82.5%; up to 2000: 85.25%).  So the conjecture here is that as the number of the amicable numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches 100%. [corrected by Paul Zimmermann, Feb 05-06 2019]
Among the 1947667 pairs up to 19 digits from Sergei Chernykh's database, there are 1872573 pairs with m+n divisible by ten, thus about 96.14%. - Paul Zimmermann, Feb 07 2019

			The sum of 6232 and 6368 is divisible by ten, thus the (6232, 6368) amicable pair belongs to the sequence. On the other hand, the (220, 284) amicable pair does not qualify since its sum is 504.

Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994, pp. 55-58.
Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chappman and HALL/CRC, 2003, pp. 67-69.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sergei Chernykh, Amicable pairs list.
Zoltan Galantai, List of the amicable pairs where the sum divisible by ten smaller and larger amicable numbers; sums (the first 4406 pairs).
Zoltan Galantai, List of the first 5001 amicable pairs with their sums denoting whether the sum is divisible by ten or not.
Eric Weisstein's World of Mathematics, Amicable Pair.
Eric Weisstein's World of Mathematics, Sociable Numbers.

Cf. A002025, A002046, A063990, A180164, A259180, A259933, A291550.

lista(nn) = {for (n=1, nn, spd = sigma(n)-n; if ((spd > n) && (sigma(spd)-spd == n) && !((n + spd) % 10), print1(n, ", ", spd, ", ")););} \\ Michel Marcus, Aug 26 2017

A307962 Lesser of coreful amicable numbers pair: numbers (m, n) such that csigma(m) = csigma(n) = m + n, where csigma(n) is the sum of the coreful divisors of n (A057723).

Original entry on oeis.org

1718200, 4818880, 5154600, 12027400, 14456640, 22336600, 29209400, 32645800, 33732160, 36082200, 39518600, 49827800, 53264200, 62645440, 63573400, 67009800, 70446200, 73882600, 80755400, 81920960, 87628200, 91064600, 91558720, 97937400, 101196480, 101373800

Offset: 1

Amiram Eldar, May 08 2019

nonn

The larger counterparts are in A307963.

If (m, n) is an amicable pair (A259180), then the pair (m*k, n*k) with k=rad(m*n) is a coreful amicable pair (rad(i)=A007947(i) is the squarefree kernel of i), and so are all the pairs (m*k*s, n*k*s) where s is a squarefree number with gcd(s, k) = 1. Proof: k = rad(m*n) = rad(m)*rad(n)/rad(gcd(m,n)), csigma(m*k) = csigma(m*rad(m)*j) where j = rad(n)/rad(gcd(m,n)) is squarefree and coprime to m*rad(m), so csigma(m*k) = j * csigma(m*rad(m)) = j * rad(m)* sigma(m) = rad(m)*rad(n)/rad(gcd(m,n)) * sigma(m) = rad(m)*rad(n)/rad(gcd(m,n)) * (n+m) = k *(n+m) = csigma(n*k).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007947, A057723, A063990, A259180, A307958, A307959, A307963.

f[p_,e_] := (p^(e+1)-1)/(p-1)-1; csigma[1]=1; csigma[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[m = csigma[n] - n; If[m > n && csigma[m] - m == n, AppendTo[s, n]], {n, 1, 10^8}]; s

Wrong terms corrected by Amiram Eldar, Dec 02 2019

A383483 Numbers k such that k = sigma(m)-m where m = sigma(3k)-3k.

Original entry on oeis.org

3, 15, 5919, 118719, 179871, 33750303

Offset: 1

S. I. Dimitrov, Apr 28 2025

S. I. Dimitrov introduced the notion of (alpha, beta)-amicable pairs.

			For alpha=1, beta=3 we have (3, 4), (15, 33), (5919, 7905).
Here (3, 4) is such a pair because 3=sigma(4)-4 and 4=sigma(3*3)-3*3.

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A063990, A259180, A383239.

isok(k) = my(m = sigma(3*k) - 3*k); if (m>0, sigma(m) - m == k); \\ Michel Marcus, Apr 28 2025

We say that the numbers m and n form an (alpha, beta)-amicable pair if sigma(alpha*n)-alpha*n=m and sigma(beta*m)-beta*m=n, where alpha and beta are positive integers, and sigma(n) is the sum of the divisors of n.

a(4)-a(6) from Michel Marcus, Apr 28 2025

A385586 Primes p such that there exists prime q < p such that sigma(p+1) = sigma(q+1) = p + q.

Original entry on oeis.org

37, 34687, 65587, 2089951, 8161477, 8340613, 18927067, 25855567, 64346413, 95150203, 238973101, 257658061, 277743397, 322210813, 349883707, 578403913, 704710543, 1121445337, 1654635937, 1741780693, 1804380007, 1963734061, 2346701941, 2360966173, 2720420707, 3232299517, 4343250181, 4925742973, 8085909913, 9044601133

Offset: 1

S. I. Dimitrov, Jul 03 2025

nonn

The primes p and q form a P(1, 1)-amicable pair. See Dimitrov link.

			(23, 37) is such a pair because sigma(23+1)=sigma(37+1) = 23 + 37.

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A000040, A063990, A008333, A259180, A385718.

Subequence of A385705.

upto(n) = {
    res = List();
    forprime(p = 2, n,
        s = sigma(p+1);
        q = s - p;
        if(q < p && isprime(q) && sigma(q+1) == s,
            print([p, q]);
            listput(res, p);
        );
    ); res
} \\ David A. Corneth, Jul 03 2025

More terms from David A. Corneth, Jul 03 2025

A000173 Unitary-sociable numbers (smallest member of each cycle).

Original entry on oeis.org

30, 1482, 2418, 24180, 35238, 263820, 395730, 473298, 698130, 763620, 2212026, 2233554, 172459210, 209524210, 341354790, 384121920, 525150234, 530946330, 582129630, 1799281330, 2069510520, 2514290520, 3344596854, 5251502340

Offset: 1

N. J. A. Sloane

If n=product p_i^a_i, d=product p_i^c_i is a unitary divisor of n if each c_i is 0 or a_i.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Sect. B7.

Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 01 2006, Table of n, a(n) for n = 1..44
David Moews, Perfect, amicable and sociable numbers.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
Eric Weisstein's World of Mathematics, Sociable Numbers.

Cf. A034444, A003416, A097323.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 01 2006

A004607 Infinitary sociable numbers (smallest member of cycle).

Original entry on oeis.org

1026, 10098, 10260, 12420, 41800, 45696, 100980, 241824, 448800, 512946, 685440, 830568, 4938136, 6732000, 9424800, 12647808, 13959680, 14958944, 17878998, 25581600, 28158165, 32440716, 36072320, 55204500, 74062944

Offset: 1

N. J. A. Sloane

If n = Product p_i^a_i, d = Product p_i^c_i is an infinitary divisor of n if each c_i has a zero bit in its binary representation everywhere that the corresponding a_i does.
From Amiram Eldar, Mar 25 2023: (Start)
Analogous to A003416 with the sum of the aliquot infinitary divisors function (A126168) instead of A001065.
Only cycles of length greater than 2 are here. Cycles of length 1 correspond to infinitary perfect numbers (A007357), and cycles of length 2 correspond to infinitary amicable pairs (A126169 and A126170).
The corresponding cycles are of lengths 4, 4, 4, 6, 4, 4, 4, 4, 11, 6, 4, 6, 4, 11, 6, 23, 4, 4, 85, 4, 4, 4, 4, 4, 4, ...
It is conjectured that there are no missing terms in the data, but it was not proven. For example, it is not known that the infinitary aliquot sequence that starts at 840 does not reach 840 again (see A361421). (End)

Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp. 54 (189) (1990) 395-411.
J. O. M. Pedersen, Tables of Aliquot Cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]

Cf. A001065, A003416, A007357, A126168, A126169, A126170, A361421.

A061193 Number of digits in n-th even perfect number (A000396).

Original entry on oeis.org

1, 2, 3, 4, 8, 10, 12, 19, 37, 54, 65, 77, 314, 366, 770, 1327, 1373, 1937, 2561, 2663, 5834, 5985, 6751, 12003, 13066, 13973, 26790, 51924, 66530, 79502, 130100, 455663, 517430, 757263, 841842, 1791864, 1819050, 4197919, 8107892, 12640858, 14471465, 15632458, 18304103, 19616714, 22370543

Offset: 1

N. J. A. Sloane

The next known values following a(48) are 44677235, 46498850, and 49724095, but these may not be the next terms. [Updated by M. F. Hasler, Nov 28 2017, Ivan Panchenko, Apr 07 2018, Apr 17 2018, Amiram Eldar, Oct 16 2024]

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.
Martin Gardner, Mathematical Magic Show, Alfred A. Knopf, 1977, p. 165.
Paul Hoffman, Archimedes' Revenge, Penguin, 1988, p. 11.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 244-245.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
Donald D. Spencer, Key Dates in Number Theory History, Camelot Pub. Co., 1995, p. 80.

Ivan Panchenko, Table of n, a(n) for n = 1..48 [Updated by _Ivan Panchenko_, Apr 17 2018, _Amiram Eldar_, Oct 16 2024]
A. Byerley, Relationship Between Mersenne Primes and Perfect Numbers.
Chris K. Caldwell, Table of Known Mersenne Primes.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
H. J. Smith, Mersenne Primes. [broken link]

Cf. A000043, A000396, A007524, A055642.

Table[n=MersennePrimeExponent@k;IntegerLength[2^(n-1)(2^n-1)],{k,45}] (* Giorgos Kalogeropoulos, Sep 03 2020 *)
Array[IntegerLength@*PerfectNumber, 18] (* Giorgos Kalogeropoulos, Sep 03 2020 *)

apply( p->(2*p-1)*log(2)\log(10)+1, A000043) \\ where A000043 is the vector of the known Mersenne primes. - M. F. Hasler, Nov 28 2017

a(n) = ceiling((2*A000043(n)-1)*A007524), with A000043 = Mersenne prime exponents, A007524 = log_10(2). - M. F. Hasler, Nov 28 2017

This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Lekraj Beedassy, May 30 2001
More terms from Harry J. Smith, Apr 16 2003
Entry revised by N. J. A. Sloane, Jun 10 2012
a(39) through a(45) from M. F. Hasler, Nov 28 2017

A066873 Number of amicable pairs where smaller term of the pair is less than 10^n.

Original entry on oeis.org

0, 0, 1, 5, 13, 42, 108, 236, 586, 1427, 3340, 7642, 17519, 39374, 87102, 190775, 415523, 901312, 1947667, 4197267

Offset: 1

Shyam Sunder Gupta, Jan 21 2002

			There are 13 pairs with smaller term of the pair < 10^5, so a(5) = 13.

Sergei Chernykh, Amicable pairs list.
Shyam Sunder Gupta, Amicable Numbers.
J. O. M. Pedersen, Known Amicable Pairs. [Broken link; WayBackMachine archive.]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link; WayBackMachine archive.]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Index entries for sequences related to amicable numbers

Cf. A063990, A259180 (amicable pairs).

Edited by Klaus Brockhaus, May 31 2003
a(15)-a(18) (from Sergei Chernykh's site) added by Amiram Eldar, Aug 19 2017
a(19) (from Sergei Chernykh's site) added by Amiram Eldar, Apr 05 2019
a(20) (from Sergei Chernykh's site) added by Amiram Eldar, Dec 13 2020

cp's OEIS Frontend

A262622 Amicable pairs of even numbers.

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A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.

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A291422 List of pairs of amicable numbers (m,n) where the sum of the pair is divisible by 10.

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A307962 Lesser of coreful amicable numbers pair: numbers (m, n) such that csigma(m) = csigma(n) = m + n, where csigma(n) is the sum of the coreful divisors of n (A057723).

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A383483 Numbers k such that k = sigma(m)-m where m = sigma(3*k)-3*k.

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A385586 Primes p such that there exists prime q < p such that sigma(p+1) = sigma(q+1) = p + q.

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A000173 Unitary-sociable numbers (smallest member of each cycle).

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A004607 Infinitary sociable numbers (smallest member of cycle).

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A383483 Numbers k such that k = sigma(m)-m where m = sigma(3k)-3k.