A262625 -id:A262625 - cp's OEIS frontend

A262623 Amicable pairs of odd numbers.

12285, 14595, 67095, 71145, 69615, 87633, 100485, 124155, 122265, 139815, 522405, 525915, 802725, 863835, 947835, 1125765, 1175265, 1438983, 1280565, 1340235, 1358595, 1486845, 1798875, 1870245, 4482765, 5120595, 5357625, 5684679, 5730615, 6088905, 6377175, 6680025, 8619765, 9627915, 9071685, 9498555, 9206925, 10791795

Offset: 1

Omar E. Pol, Nov 09 2015

If there are no amicable pairs whose members have distinct parity then this is also the odd terms of A259180.

First differs from A262625 at a(4).

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

Amiram Eldar, Table of n, a(n) for n = 1..20000
Sergei Chernykh, Amicable pairs list

Cf. A002025, A002046, A005408, A063990, A259180, A262622, A262625.

lista(nn) = {forstep(n=1, nn, 2, m = sigma(n)-n; if ((m > n) && (n==sigma(m)-m), print1(n, ", ", m, ", ")););} \\ Michel Marcus, Nov 14 2015

A262624 Even amicable numbers.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 02 2015

nonn

Even numbers that are also amicable numbers.
Intersection of A005843 and A063990.
The first time a pair (x, y) of even amicable numbers ordered by its first element is not adjacent is x = 63020, y = 76084 which correspond to a(15) and a(18), respectively.

Cf. A002025, A002046, A005843, A063990, A259180, A259933, A262625.

t(n)=sigma(n)-n;
is(n)={local(a); a=t(n); a<>n && t(a)==n};
for(n=1, 1e6, if( n%2 == 0 && is(n), print1(n", "))) \\ Altug Alkan, Oct 16 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.

A358320 Least odd number m such that m*2^n is a perfect, amicable or sociable number, and -1 if no such number exists.

Original entry on oeis.org

12285, 3, 7, 779, 31, 37, 127, 651, 2927269, 93, 25329329, 7230607, 8191, 66445153, 7613527, 18431675687, 131071, 264003743, 524287, 59592560831, 949755039781

Offset: 0

Jean-Marc Rebert, Nov 09 2022

For n in {1,2,4,6,12,16,18}, a(n)*2^n is a perfect number. See A090748.
For n in {0,3,5,8,10,11,13,14,15,17,19}, a(n)*2^n is an amicable number.
For n in {7,9} a(n)*2^n is a sociable number of order 28.
That is, h_k(m*2^n) = m*2^n for some k > 0, where h_{k+1}(n) = h_k(h(n)) and h(n) = A001065(n), the sum of aliquot parts of n. - Charles R Greathouse IV, Nov 09 2022
Least m such that m*2^n is in A347770. - Charles R Greathouse IV, Nov 09 2022

			a(1) = 3, because 3 is an odd number and 3 * 2^1 = 6 is a perfect number, and no lesser number has this property.

BOINC, Amicable Numbers
HandWiki, List of perfect numbers
David Moews, A list of currently known aliquot cycles of length greater than 2

Cf. A347770, A000396, A001065, A002025, A259180, A262625.

Cf. A090748, A358415.

sigmap(n)=if(n<=1, return(0)); sigma(n)-n
cycle(n,TT=28)=my(x=n,T=1); while(x>0&&T<=TT, x=sigmap(x); if(x==n, return(T)); T++)
a(n,TT=28)=my(p2n=2^n); forstep(m=1, +oo, 2, if(cycle(p2n*m,TT), return(m)))

a(0), a(15)-a(20) from Jean-Marc Rebert, Nov 17 2022

A358022 Least odd number m such that m*2^n is an amicable number, and -1 if no such number exists.

Original entry on oeis.org

12285, 605, 55, 779, 1081, 37, 119957, 73153, 2927269, 239, 25329329, 7230607, 964119281, 66445153, 7613527, 18431675687, 328796066369, 264003743, 11298797322497, 59592560831, 949755039781, 2759891672513

Offset: 0

Michel Marcus, Nov 17 2022

Similar to A358320 but restricted to amicable numbers.

Sergei Chernykh, Amicable Numbers.

Cf. A001065, A002025, A259180, A262625, A358320.

f(m) = if (m, sigma(m)-m, 0);
fpm(p, m) = (2*p-1)*sigma(m) - p*m;
a(n) = my(p=2^n); forstep(m=1, +oo, 2, my(x=fpm(p,m)); if ((x!=p*m) && (f(x) == p*m), return(m)));

a(12)-a(22) were calculated using Chernykh's database by Amiram Eldar, Nov 17 2022

A358546 Least odd number m such that m mod 3 > 0 and m*3^n is an amicable number, and -1 if no such number exists.

Original entry on oeis.org

5480828320492525, 4865, 7735, 455, 131285, 849355, 11689795, 286385, 187047685, 104255, 32851039955, 2085985, 47942199242945, 189296520259, 349700961302721360788238344333849, 580068028631, 50392682631679406080371010751466781

Offset: 0

Jean-Marc Rebert, Nov 21 2022

If a(n) > -1 then a(n)*3^n is the least amicable number k such that A007949(k) = n.

			a(1) = 4865, because 4865 is an odd number and 4865 % 3 > 0 and 4865 * 3 = 14595 is an amicable number, and no lesser number has this property.

BOINC, Amicable Numbers

Cf. A347770, A000396, A001065, A002025, A259180, A262625.

Cf. A090748, A358415, A358320, A358022.

sigmap(k)=if(k,sigma(k)-k,0)
cycle(k, TT=2)=my(x=k, T=1); while(x>0&&T<=TT, x=sigmap(x); if(x==k, return(T)); T++)
a(n, TT=2)=my(p3n=3^n); forstep(m=1, +oo, 2, if(m%3&&cycle(p3n*m, TT)==2, return(m)))

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

A262623 Amicable pairs of odd numbers.

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A262624 Even amicable 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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A358320 Least odd number m such that m*2^n is a perfect, amicable or sociable number, and -1 if no such number exists.

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A358022 Least odd number m such that m*2^n is an amicable number, and -1 if no such number exists.

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A358546 Least odd number m such that m mod 3 > 0 and m*3^n is an amicable number, and -1 if no such number exists.

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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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