A262623 -id:A262623 - 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

A354070 Lesser of an amicable pair in which both members are divisible only by primes congruent to 3 (mod 4).

Original entry on oeis.org

294706414233, 518129600373, 749347913853, 920163589191, 1692477265941, 2808347861781, 3959417614383, 4400950312143, 9190625896683, 10694894578137, 12615883061859, 15028451404659, 18971047742031, 21981625463259, 29768959571967, 37423211019579, 54939420064683, 69202873206621

Offset: 1

Amiram Eldar, May 16 2022

nonn

Since the factorization of numbers that are divisible only by primes congruent to 3 (mod 4) is the same also in Gaussian integers, these pairs are also Gaussian amicable pairs.
There are 4197267 lesser members of amicable pairs below 10^20 and only 1565 are in this sequence.
The least pair, (294706414233, 305961592167), was discovered by Herman J. J. te Riele in 1995.
The larger counterparts are in A354071.

			294706414233 is a term since (294706414233, 305961592167) is an amicable pair: A001065(294706414233) = 305961592167 and A001065(305961592167) = 294706414233, 294706414233 = 3^4 * 7^2 * 11 * 19 * 47 * 7559, and 3, 7, 11, 19, 47 and 7559 are all congruent to 3 (mod 4), and 305961592167 = 3^4 * 7 * 11 * 19 * 971 * 2659, and 3, 7, 11, 19, 971 and 2659 are all congruent to 3 (mod 4).

Amiram Eldar, Table of n, a(n) for n = 1..1565
Ranthony Ashley Clark, Gaussian Amicable Pairs, Thesis, Eastern Kentucky University, 2013.
Patrick Costello and Ranthony Clark, Gaussian Amicable Pairs: "Friendly Imaginary Numbers", 2013.
Patrick Costello and Ranthony A. C. Edmonds, Gaussian Amicable Pairs, Missouri Journal of Mathematical Sciences, Vol. 30, No. 2 (2018), pp. 107-116.
Wikipedia, Gaussian integer.

Subsequence of A002025 and A004614.

Cf. A001065, A063990, A262623, A262625, A354071.

A354071 Larger of an amicable pair in which both members are divisible only by primes congruent to 3 (mod 4).

Original entry on oeis.org

305961592167, 523630799307, 758052380547, 964086778809, 1697959925739, 2961402044139, 4049489137617, 4475588004657, 9309948700437, 10759267751463, 12799047697821, 15133576811661, 21200708842929, 22067361672741, 30807498770433, 38260957786821, 56250902008917, 70669851785379

Offset: 1

Amiram Eldar, May 16 2022

nonn

The terms are ordered according to their lesser counterparts (A354070).

See A354070 for more details.

Amiram Eldar, Table of n, a(n) for n = 1..1565

Subsequence of A002046 and A004614.

Cf. A063990, A262623, A262625, A354070.

A328256 Amicable pairs of cyclops numbers.

Original entry on oeis.org

1280565, 1340235, 71526069316, 75257076284, 1453520375775, 1561230417825, 1568650892445, 1995690781539, 2714480497936, 2854320218864, 5776910172896, 5864460215584, 5818350517628, 6516910297732, 6111770382135, 6139270339785, 9176850735616, 9194440569344, 114552504952875, 123277906567125

Offset: 1

Omar E. Pol, Oct 09 2019

Amicable pairs (A259180) where both members are cyclops numbers (A134808).
Is this sequence finite? What is the largest known pair?
Up to 2^64 there are 439 pairs, the largest of them is {9591988390446931328, 9596251990981497472} (Eldar).

			{1280565, 1340235} is the first amicable pair where both members are cyclops numbers, so a(1) = 1280565 and a(2) = 1340235.

Amiram Eldar, Table of n, a(n) for n = 1..878
OEIS Wiki, Cyclops Numbers

Subsequence of A259180.

Cf. A002025, A002046, A134808, A262622, A262623.

More terms from Amiram Eldar, Oct 09 2019

A360140 Odd amicable pairs with only one member divisible by 3.

Original entry on oeis.org

445953248528881275, 659008669204392325, 748174019876835825, 906104451346869775, 1097581690986390225, 1615281291559017775, 1281431098689616875, 1769164614201263125, 1968382462511781225, 2982869282783783575, 1993991197249826775, 2901232579265245225, 2247817805416685775, 2726235257034514225

Offset: 1

Zoltan Galantai, Jan 26 2023

nonn

In these pairs only the smaller number is divisible by 3 and both the smaller and the larger number are divisible by 5, 7, and 25. These pairs prove that there are odd amicable pairs where only one of the numbers is divisible by 3.

			(445953248528881275, 659008669204392325) belongs to the sequence since the smaller number is divisible by 3 and the larger is not. The (12285, 14595) pair is not a part of the sequence, since both of its members are divisible by 3.

Richard K. Guy, Unsolved Problems in Number Theory. Springer, 2004, page 88.

S. Battiato and W. Borho, Are There Odd Amicable Numbers Not Divisible by Three, Mathematics of Computation, Apr., 1988, Vol. 50, No. 182, pp. 633-637.
S. Chernykh, Amicable pairs where the two members have different smallest prime factors, January 30th, 2016.
Zoltan Galantai, List of 31 known pairs

Cf. A259180, A002025, A002046, A262623.

A360054 Number of odd amicable pairs where the smaller term of the pair is less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 3, 8, 21, 55, 154, 412, 1088, 2632, 6532, 15371, 35218, 79982, 180061, 402560, 894404, 1975742

Offset: 1

Zoltan Galantai, Jan 23 2023

The list starts with n=1.

Comparing with the numbers of even amicable pairs in A066873, up to 10^4, the proportion of odd amicable pairs is 0%; up to 10^5 it is 23% and up to 10^10 is 28.9%. Up to 10^15, it is 40.4% and up to 10^19 this percentage is 45.9%. It is possible that this trend holds true for more amicable pairs, and thus most amicable number pairs are odd.

Song Y. Yan, Perfect, Amicable and Sociable Numbers. A Computational Approach, World Scientific, 1996, pages 151 - 153.

Sergei Chernykh, Amicable pairs list
Zoltan Galantai, Number of odd amicable pairs where the smaller term of the pair is less than 10^20

Cf. A066873, A259180, A262623, A262622, A262625, A005408.

cp's OEIS Frontend

A262622 Amicable pairs of even numbers.

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A354070 Lesser of an amicable pair in which both members are divisible only by primes congruent to 3 (mod 4).

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A354071 Larger of an amicable pair in which both members are divisible only by primes congruent to 3 (mod 4).

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A328256 Amicable pairs of cyclops numbers.

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A360140 Odd amicable pairs with only one member divisible by 3.

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A360054 Number of odd amicable pairs where the smaller term of the pair is less than 10^n.

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