A064116 Composite numbers whose sum of aliquot divisors as well as product of aliquot divisors is a perfect square.

12, 75, 76, 124, 147, 153, 176, 243, 332, 363, 477, 507, 524, 575, 688, 867, 892, 963, 1075, 1083, 1421, 1532, 1573, 1587, 1611, 1916, 2032, 2075, 2224, 2299, 2401, 2421, 2523, 2572, 2883, 2891, 3100, 3479, 3776, 3888, 4107, 4336, 4527, 4961, 4975, 5043

Offset: 1

Author

Shyam Sunder Gupta, Sep 09 2001

Keywords

• base
• easy
• nonn

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Crossrefs

Intersection of A048699 and A064499.

Cf. A001065, A007956.

Programs

Mathematica

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} ]
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 *)

PARI

isok(k) = { my(s=sigma(k) - k); s>1 && issquare(s) && issquare(vecprod(divisors(k)[1..-2])) } \\ Harry J. Smith, Sep 07 2009

Extensions

More terms from Robert G. Wilson v, Oct 05 2001