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
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)
Programs
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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