A384255 Integers k such that there exists an integer 0<m<k such that sigma(m)^2 + sigma(k)^2 = 2*(m^2+k^2).

2, 21, 27, 123, 175, 2133, 2187, 6093, 340917, 504309, 1594323, 1895841, 5308415, 23006577, 62188641

Offset

1,1

Comments

The most interesting question that arises here is whether there exist such pairs for which sigma(m) = sigma(k), which would imply that sigma(m)^2 = sigma(k)^2 = m^2+k^2. None have been found for m < k <= 10^8.

Links

Table of n, a(n) for n=1..15.

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

Example

(13, 21) is such a pair because sigma(13)^2 + sigma(21)^2 = 14^2 + 32^2 = 2*(13^2+21^2).

Mathematica

f[n_]:=f[n]=DivisorSigma[1, n]^2-2*n^2; lst={}; Do[AppendTo[lst, f@k]; If[MemberQ[lst, -f@k], Print@k], {k, 10000}] (* Giorgos Kalogeropoulos, May 29 2025 *)

Prog

(PARI) isok(k) = for(m=1, k-1, if (sigma(m)^2 + sigma(k)^2 == 2*(m^2+k^2), return(1))); \\ Michel Marcus, May 23 2025

Crossrefs

Cf. A063990, A259180, A383484, A383714

Sequence in context: A079907 A342727 A075715 * A326735 A294377 A071761

Adjacent sequences: A384252 A384253 A384254 * A384256 A384257 A384258

Keyword

nonn,hard,more

Author

S. I. Dimitrov, May 23 2025

Extensions

a(9)-a(15) from Giorgos Kalogeropoulos, May 29 2025

Status

approved