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

1,1

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.

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A229072 A261983 A074896 * A049198 A146303 A344999

Adjacent sequences: A015710 A015711 A015712 * A015714 A015715 A015716

Keyword

nonn

Author

Robert G. Wilson v

Status

approved