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%I #29 Dec 09 2024 15:25:31
%S 12,75,76,124,147,153,176,243,332,363,477,507,524,575,688,867,892,963,
%T 1075,1083,1421,1532,1573,1587,1611,1916,2032,2075,2224,2299,2401,
%U 2421,2523,2572,2883,2891,3100,3479,3776,3888,4107,4336,4527,4961,4975,5043
%N Composite numbers whose sum of aliquot divisors as well as product of aliquot divisors is a perfect square.
%H Amiram Eldar, <a href="/A064116/b064116.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%e 75 is a term because the sum of the aliquot divisors of 75 = 1 + 3 + 5 + 15 + 25 = 49 = 7^2 and the product of the aliquot divisors of 75 = 1*3*5*15*25 = 75^2.
%t Do[d = Delete[ Divisors[n], -1]; If[ !PrimeQ[n] && IntegerQ[ Sqrt[ Apply[ Plus, d]]] && IntegerQ[ Sqrt[ Apply[ Times, d]]], Print[n]], {n, 2, 10^4} ]
%t spsQ[n_]:=Module[{d=Most[Divisors[n]]},CompositeQ[n]&&AllTrue[{Sqrt[ Total[ d]],Sqrt[Times@@d]},IntegerQ]]; Select[Range[5100],spsQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Feb 14 2018 *)
%o (PARI) isok(k) = { my(s=sigma(k) - k); s>1 && issquare(s) && issquare(vecprod(divisors(k)[1..-2])) } \\ _Harry J. Smith_, Sep 07 2009
%Y Intersection of A048699 and A064499.
%Y Cf. A001065, A007956.
%K base,easy,nonn
%O 1,1
%A _Shyam Sunder Gupta_, Sep 09 2001
%E More terms from _Robert G. Wilson v_, Oct 05 2001