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%I #27 May 04 2025 23:44:16
%S 1,6,9,28,216,360,496,2016,2401,8128,16758,182520,884736,1022112,
%T 1352328,1571328,1935360,2678400,33550336,54758400,101382400,
%U 119533176,136808280,163298502,198288000,618591192,691022088,782481673,796663296,1137067008,1275418369,1303102080
%N Integers k such that sigma(k)/k - 1 is a rational square.
%H Michel Marcus, <a href="/A383482/b383482.txt">Table of n, a(n) for n = 1..40</a>
%e 6 is a term because sigma(6)/6 - 1 = 2 - 1 = 1, a square; like for all perfect numbers.
%e 9 is a term because sigma(9)/9 - 1 = 13/9 - 1 = 4/9, a square.
%t q[k_] := And @@ IntegerQ /@ Sqrt[NumeratorDenominator[DivisorSigma[-1, k] - 1]]; Select[Range[2*10^6], q] (* _Amiram Eldar_, Apr 28 2025 *)
%o (PARI) isok(k) = issquare(sigma(k)/k - 1);
%Y Cf. A000203 (sigma), A069070.
%Y Subsequences: A000396 (perfect numbers), A046060 (5-multiperfect numbers), A381321.
%Y Cf. A218404 (for those terms with sigma(x)/x = 13/4).
%K nonn
%O 1,2
%A _Michel Marcus_, Apr 28 2025
%E a(30)-a(32) from _Jinyuan Wang_, Apr 28 2025