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

Author

Amiram Eldar, May 16 2022

Keywords

• nonn

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.

Examples

			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).
		

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.

Crossrefs

Subsequence of A002025 and A004614.

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