A383714 Integers k such that there exists an integer 0<m<k such that m*sigma(m)^2 + k*sigma(k)^2 = (m+k)^3.

21, 231, 284, 1210, 2499, 2924, 5564, 6368, 10856, 14595, 18416, 66992, 71145, 76084, 87633, 88730, 123152, 124155, 139815, 153176, 168730, 176336, 180848, 202444, 203432, 293986, 365084, 389924, 399592, 430402, 455344, 486178, 514736, 525915, 652664, 669688, 686072, 691256, 712216

Offset

1,1

Comments

The numbers m and k form a WPM(2)-amicable pair (WPM = weighted power mean). See Dimitrov link.

Links

Max Alekseyev, Table of n, a(n) for n = 1..105

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

Example

(7, 21) is such a pair because 7*sigma(7)^2 + 21*sigma(21)^2 = 7*8^2 + 21*32^2 = (7+21)^3.

Prog

(PARI) isok(k)= for (m=1, k-1, if (m*sigma(m)^2 + k*sigma(k)^2 == (m+k)^3, return(m))); \\ Michel Marcus, May 15 2025
(PARI) is_a383714(k) = if(k<=1, return(0)); my(ks=k*sigma(k)^2); fordiv(k^3 - ks, m, if(m>=k, break); if(m*sigma(m)^2 + ks == (m+k)^3, return(m) ) ); 0; \\ Max Alekseyev, Nov 15 2025

Crossrefs

Cf. A002046 (a subsequence), A063990, A259180, A383239, A383483, A383484.

Sequence in context: A162360 A161899 A220721 * A320588 A020267 A188354

Adjacent sequences: A383711 A383712 A383713 * A383715 A383716 A383717

Keyword

nonn

Author

S. I. Dimitrov, May 14 2025

Extensions

a(3)-a(16) from Michel Marcus, May 15 2025

Terms a(17) onward from Max Alekseyev, Nov 15 2025

Status

approved