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A318585 Number of integer partitions of n whose sum of reciprocals squared is an integer. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

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)

LINKS

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

EXAMPLE

The a(26) = 7 integer partitions:

  (6332222222)

  (44442221111)

  (63322211111111)

  (22222222222211)

  (222222221111111111)

  (2222111111111111111111)

  (11111111111111111111111111)

MATHEMATICA

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

CROSSREFS

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

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

Sequence in context: A105390 A013941 A061798 * A029241 A226749 A277090

Adjacent sequences:  A318582 A318583 A318584 * A318586 A318587 A318588

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 29 2018

EXTENSIONS

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

STATUS

approved

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Last modified February 27 13:18 EST 2020. Contains 332306 sequences. (Running on oeis4.)