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A064180 Composite numbers k such that the sum of the proper divisors of k not including 1, (Chowla's function, A048050) and their product (A007956) are both perfect squares.

%I A064180 #13 Sep 08 2022 08:45:04
%S A064180 117,208,292,320,475,539,549,567,873,964,1737,2107,2692,2997,3573,
%T A064180 3904,4477,4802,5275,5284,5968,6057,7267,7488,7492,9189,9457,9475,
%U A064180 10084,10377,11072,11728,11737,12717,13769,14373,14692,16219,16399,17397
%N A064180 Composite numbers k such that the sum of the proper divisors of k not including 1, (Chowla's function, A048050) and their product (A007956) are both perfect squares.
%H A064180 Amiram Eldar, <a href="/A064180/b064180.txt">Table of n, a(n) for n = 1..10000</a>
%e A064180 117 is in the sequence because the divisors of 117 are 1, 3, 9, 13, 39 and 117. Being squarefree itself, the product of divisors is a perfect square. The sum of the divisors in question, 3+9+13+39 = 64 and it is a perfect square.
%t A064180 Select[ Range[2, 25000], IntegerQ[ Sqrt[ Apply[ Plus, Delete[ Divisors[ # ], -1]] - 1]] && IntegerQ[ Sqrt[ Apply[ Times, Delete[ Divisors[ # ], -1]]]] && ! PrimeQ[ # ] & ]
%t A064180 aQ[n_] := CompositeQ[n] && IntegerQ[Sqrt[n^(DivisorSigma[0, n]/2 - 1)]] && IntegerQ[Sqrt[DivisorSigma[1, n] - 1 - n]]; Select[Range[18000], aQ] (* _Amiram Eldar_, Jul 03 2019 *)
%o A064180 (Magma) [k:k in [1..18000]| not IsPrime(k) and IsSquare((&+Divisors(k))-1-k) and IsSquare((&*Divisors(k))/k) ]; // _Marius A. Burtea_, Jul 03 2019
%Y A064180 Cf. A000290, A048050, A007956.
%K A064180 easy,nonn
%O A064180 1,1
%A A064180 _Robert G. Wilson v_, Oct 14 2001