A384706 Integers y such that there exists an integer 0 < x < y such that y/sigma(x) + x/sigma(y) = 1.

14, 20, 42, 54, 62, 88, 99, 108, 114, 124, 126, 132, 189, 195, 204, 210, 220, 238, 252, 254, 272, 284, 328, 340, 385, 414, 420, 432, 455, 464, 468, 495, 508, 528, 560, 572, 608, 621, 630, 663, 693, 748, 828, 837, 870, 888, 1008, 1089, 1136, 1192, 1197, 1210, 1288, 1416, 1422, 1440

Offset

1,1

Comments

Let sigma be the usual sum-of-divisors function. We say that x and y form a cross-harmonious pair if y/sigma(x) + x/sigma(y) = 1. An amicable pair forms a cross-harmonious pair, so the larger member of an amicable pair A002046 is a term of this sequence.

If a < b are Mersenne exponents (A000043) then 2^(a-1) * (2^b - 1) is a term, as it and 2^(b-1) * (2^a - 1) are a cross-harmonious pair. - Robert Israel, Jul 07 2025

Links

Robert Israel, Table of n, a(n) for n = 1..500

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

Example

12 and 14 form a cross-harmonious pair since 14/sigma(12) + 12/sigma(14) = 14/28 + 12/24 = 1.

Maple

N:= 10000: # for terms <= N
S:= map(numtheory:-sigma, [$1..N]):
filter:= proc(x) ormap(y -> y/S[x]+x/S[y]=1, [$1..x-1]) end proc:
select(filter, [$1..N]); # Robert Israel, Jul 07 2025

Prog

(PARI) isok(y) = for (x=1, y-1, if (y/sigma(x) + x/sigma(y) == 1, return(x)); ); \\ Michel Marcus, Jun 09 2025

Crossrefs

Cf. A000043, A000203, A000396, A002025, A002046, A253535, A253534.

Sequence in context: A054017 A129216 A259536 * A067796 A395421 A108420

Adjacent sequences: A384703 A384704 A384705 * A384707 A384708 A384709

Keyword

nonn

Author

S. I. Dimitrov, Jun 07 2025

Status

approved