A102924 Real part of Gaussian amicable numbers in order of increasing magnitude. See A102925 for the imaginary part.

-1105, -1895, -2639, -3235, -3433, -3970, -4694, -3549, -766, -4478, -6880, 5356, -6468, 8008, 4232, -8547

Offset

1,1

Comments

For a Gaussian integer z, let the sum of the proper divisors be denoted by s(z) = sigma(z)-z, where sigma(z) is sum of the divisors of z, as defined by Spira for Gaussian integers. Then z is an amicable Gaussian number if z and s(z) are different and z = s(s(z)). The smallest Gaussian amicable number in the first quadrant is 8008+3960i.

Links

Table of n, a(n) for n=1..16.

R. Spira, The Complex Sum Of Divisors, American Mathematical Monthly, 1961 Vol. 68, pp. 120-124.

Eric Weisstein's World of Mathematics, Amicable Pair

Example

For z=-1105+1020i, we have s(z)=-2639-1228i and s(s(z))=z.

Mathematica

s[z_Complex] := DivisorSigma[1, z]-z; nn=10000; lst={}; Do[d=a^2+b^2; If[d<nn^2, z=a+b*I; Do[If[s[s[z]]==z, AppendTo[lst, {d, z}]]; z=z*I, {4}]], {a, nn}, {b, nn}]; Re[Transpose[Sort[lst]][[2]]]

Crossrefs

Cf. A102506 (Gaussian multiperfect numbers), A102531 (absolute Gaussian perfect numbers).

Keyword

sign,more

Author

T. D. Noe, Jan 19 2005

Status

approved

A102925 Imaginary part of Gaussian amicable numbers in order of increasing magnitude. See A102924 for the real part.

1020, 2060, -1228, 1020, -2356, 2435, 467, -4988, -6187, -5471, 4275, -6133, -5251, 3960, -8280, 4606

Offset

1,1

Links

Table of n, a(n) for n=1..16.

Mathematica

s[z_Complex] := DivisorSigma[1, z]-z; nn=10000; lst={}; Do[d=a^2+b^2; If[d<nn^2, z=a+b*I; Do[If[s[s[z]]==z, AppendTo[lst, {d, z}]]; z=z*I, {4}]], {a, nn}, {b, nn}]; Im[Transpose[Sort[lst]][[2]]]

Keyword

sign

Author

T. D. Noe, Jan 19 2005

Status

approved

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

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

Offset

1,1

Comments

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.

Links

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.

Example

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).

Crossrefs

Subsequence of A002025 and A004614.

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

Keyword

nonn

Author

Amiram Eldar, May 16 2022

Status

approved