A363967 Numbers whose divisors can be partitioned into two disjoint sets whose both sums are squares.

1, 3, 9, 22, 27, 30, 40, 63, 66, 70, 81, 88, 90, 94, 115, 119, 120, 138, 153, 156, 170, 171, 174, 184, 189, 190, 198, 210, 214, 217, 232, 264, 265, 270, 280, 282, 310, 318, 322, 323, 343, 345, 357, 360, 364, 376, 382, 385, 399, 400, 414, 416, 462, 468, 472, 495, 497

Offset

1,2

Comments

If one of the two sets is empty then the term is a number whose sum of divisors is a square (A006532).

If k is a number such that (6*k)^2 is the sum of a twin prime pair (A037073), then (18*k^2)^2 - 1 is a term.

3 is the only prime term.

Links

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

Example

9 is a term since its divisors, {1, 3, 9}, can be partitioned into the two disjoint sets, {1, 3} and {9}, whose sums, 1 + 3 = 4 = 2^2 and 9 = 3^2, are both squares. 	

Mathematica

sqQ[n_] := IntegerQ[Sqrt[n]]; q[n_] := Module[{d = Divisors[n], s, p}, s = Total[d]; p = Position[Rest @ CoefficientList[Product[1 + x^i, {i, d}], x], _?(# > 0 &)] // Flatten; AnyTrue[p, sqQ[#] && sqQ[s - #] &]]; Select[Range[500], q]

Crossrefs

Cf. A000203, A000290, A037073.

Subsequence of A333911.

A006532 is a subsequence.

Similar sequences: A333677, A360694.

Sequence in context: A098980 A007647 A247182 * A318807 A063586 A131477

Adjacent sequences: A363964 A363965 A363966 * A363968 A363969 A363970

Keyword

nonn

Author

Amiram Eldar, Jun 30 2023

Status

approved