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A263928 Integers m such that sigma(m)^2 is divisible by m.

%I A263928 #33 Jul 05 2023 15:58:48
%S A263928 1,6,24,28,120,224,234,270,496,588,600,672,864,1080,1521,1638,1782,
%T A263928 2016,3724,4320,4680,5733,6048,6200,6552,7128,8128,11172,11466,15872,
%U A263928 17280,18144,18600,18620,21600,22932,26208,26460,27000,30240,32640,32760,33516,35640
%N A263928 Integers m such that sigma(m)^2 is divisible by m.
%C A263928 Previous name was: "Numbers such that the product of the sum of their divisors and the sum of the reciprocals of their divisors is an integer".
%C A263928 The multiply-perfect numbers (A007691) are a subset of this sequence.
%C A263928 This is a subsequence of A175200. - _Michel Marcus_, Nov 03 2015
%C A263928 Alternative definition: Numbers m such that Sum_{i = 1..k} (sigma(m) - d_i) / d_i is an integer, where d_i are the k divisors of m. - _Paolo P. Lava_, Mar 23 2017
%H A263928 Paolo P. Lava, <a href="/A263928/b263928.txt">Table of n, a(n) for n = 1..100</a>
%e A263928 Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Their sum is sigma(24) = 60 while the sum of their reciprocals is 1/1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/12 + 1/24 = 5/2. Finally 60 * 5/2 = 150.
%p A263928 with(numtheory): P:=proc(q) local n;
%p A263928 for n from 1 to q do if type(sigma(n)^2/n,integer) then print(n);
%p A263928 fi; od; end: P(10^6);
%t A263928 Select[Range[36000],Divisible[DivisorSigma[1,#]^2,#]&] (* _Harvey P. Dale_, Jul 05 2023 *)
%o A263928 (PARI) isok(n) = (sigma(n)^2 % n) == 0; \\ _Michel Marcus_, Nov 03 2015
%Y A263928 Cf. A000203, A007691, A017665, A017666, A175200, A263983.
%K A263928 nonn
%O A263928 1,2
%A A263928 _Paolo P. Lava_, Oct 30 2015
%E A263928 New name from _Michel Marcus_, Nov 03 2015