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A064116
Composite numbers whose sum of aliquot divisors as well as product of aliquot divisors is a perfect square.
3
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
a(2)=75 because sum of aliquot divisors of 75 = 1 + 3 + 5 + 15 + 25 = 49 = 7^2 and product of 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) pad(n)= { local(d, p=1); d=divisors(n); for (i=1, length(d) - 1, p*=d[i]); return(p) }
{ n=0; for (m=2, 10^9, s=sigma(m) - m; if (s>1 && issquare(s) && issquare(pad(m)), write("b064116.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 07 2009
CROSSREFS
Cf. A048699.
Sequence in context: A055912 A346446 A064121 * A003368 A246767 A328526
KEYWORD
base,easy,nonn
AUTHOR
Shyam Sunder Gupta, Sep 09 2001
EXTENSIONS
More terms from Robert G. Wilson v, Oct 05 2001
STATUS
approved