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.

117, 208, 292, 320, 475, 539, 549, 567, 873, 964, 1737, 2107, 2692, 2997, 3573, 3904, 4477, 4802, 5275, 5284, 5968, 6057, 7267, 7488, 7492, 9189, 9457, 9475, 10084, 10377, 11072, 11728, 11737, 12717, 13769, 14373, 14692, 16219, 16399, 17397

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

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.

Mathematica

Select[ Range[2, 25000], IntegerQ[ Sqrt[ Apply[ Plus, Delete[ Divisors[ # ], -1]] - 1]] && IntegerQ[ Sqrt[ Apply[ Times, Delete[ Divisors[ # ], -1]]]] && ! PrimeQ[ # ] & ]
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 *)

Prog

(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

Crossrefs

Cf. A000290, A048050, A007956.

Sequence in context: A272389 A201021 A112877 * A050245 A351574 A031174

Adjacent sequences: A064177 A064178 A064179 * A064181 A064182 A064183

Keyword

easy,nonn

Author

Robert G. Wilson v, Oct 14 2001

Status

approved