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

1,1

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Intersection of A048699 and A064499.

Cf. A001065, A007956.

Sequence in context: A055912 A346446 A064121 * A003368 A246767 A328526

Adjacent sequences: A064113 A064114 A064115 * A064117 A064118 A064119

Keyword

base,easy,nonn

Author

Shyam Sunder Gupta, Sep 09 2001

Extensions

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

Status

approved