A015713 Numbers m such that phi(m) * sigma(m) + k^2 is not a square for any k.

4, 9, 18, 49, 81, 98, 121, 162, 242, 361, 529, 722, 729, 961, 1058, 1458, 1849, 1922, 2209, 2401, 3481, 3698, 4418, 4489, 4802, 5041, 6241, 6561, 6889, 6962, 8978, 10082, 10609, 11449, 12482, 13122, 13778, 14641, 16129, 17161

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn

Comments

Numbers m such that A062354(m) is in A016825. - Michel Marcus, Dec 07 2018

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Crossrefs

Cf. A007814, A015710, A062354 (phi(n)*sigma(n)), A016825.

Programs

Mathematica

nonSqDiffQ[n_] := Mod[n, 4] == 2; aQ[n_] := nonSqDiffQ[ EulerPhi[n] * DivisorSigma[ 1, n]]; Select[Range[20000], aQ] (* Amiram Eldar, Dec 07 2018 *)

PARI

isok(n) = (sigma(n)*eulerphi(n) % 4) == 2; \\ Michel Marcus, Dec 07 2018

Formula

Conjecture: {4, p^(2*m), 2*p^(2*m), p = 4*k+3 is prime}. - Sean A. Irvine, Dec 06 2018

The conjecture is true. It can be proved using the multiplicative property of A062354(n), i.e., A062354(p^e) = p^(e-1)*(p^(e+1)-1), and that if m is a term then A007814(A062354(m)) = 1. - Amiram Eldar, Feb 11 2024