A318585 Number of integer partitions of n whose sum of reciprocals squared is an integer.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 12, 12, 13, 14, 16, 16, 18, 19, 21, 23, 26, 27, 29, 30, 34, 35, 39, 43, 48, 51, 55, 57, 63, 67, 74, 78, 84, 89, 99, 103, 112, 119, 132, 139, 148, 156, 170, 182, 199

Offset: 1

Author

Gus Wiseman, Aug 29 2018

Keywords

• nonn

Comments

From David A. Corneth, Sep 03 2018: (Start)

Let a valid tuple be a tuple of positive integers whose sum of reciprocals squared is an integer. Initially one only needs to consider tuples of positive integers where each element is > 1. After that some ones could be prepended to a valid tuple to find new valid tuples.

One could define a prime tuple as a valid tuple where no proper part with elements is a valid tuple. So (1) would be a prime tuple as no proper part of (1) has elements and is a valid tuple. Other examples of prime tuples are (2, 2, 2, 2) and (2, 2, 2, 3, 3, 6).

The list of distinct elements in a tuple could be whittled down by finding for each positive integer m the least sum of a prime tuple in which that integer is. For each m, that sum is at most m^3. (End)

Examples

			The a(26) = 7 integer partitions:
  (6332222222)
  (44442221111)
  (63322211111111)
  (22222222222211)
  (222222221111111111)
  (2222111111111111111111)
  (11111111111111111111111111)
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..130

Crossrefs

Cf. A000041, A002865, A051908, A058360, A289506, A289507, A316854, A316856.

Cf. A316854, A318584, A318586, A318587, A318588, A318589.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],IntegerQ[Total[#^(-2)]]&]],{n,30}]

Extensions

a(61)-a(70) from Giovanni Resta, Sep 03 2018