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