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