A366851 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n such that the sum of primes indexed by all parts greater than one is k.

1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 1, 0, 0, 1

Offset

0,43

Comments

To illustrate the definition, the sum of primes indexed by all parts greater than one of the partition (5,2,2,1) is prime(5) + prime(2) + prime(2) = 17.

Links

Table of n, a(n) for n=0..107.

Example

Triangle begins:
  1
  1
  1 0 0 1
  1 0 0 1 0 1
  1 0 0 1 0 1 1 1
  1 0 0 1 0 1 1 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 1 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 1 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 2 0 2 1 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 2 2 2 2 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 4 2 3 2 0 3 1 0 0 0 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 5 4 4 3 3 3 2 3 0 1 0 0 1 0 1
The T(8,13) = 3 partitions are: (6,1,1), (4,2,2), (3,3,2).
The T(10,17) = 4 partitions are: (7,1,1,1), (5,2,2,1), (4,4,2), (4,3,3).

Mathematica

Table[Length[Select[IntegerPartitions[n], Total[Select[Prime/@#, OddQ]]==k&]], {n, 0, 10}, {k, 0, If[n<=1, 0, Prime[n]]}]

Crossrefs

Row lengths are A055670.

Columns appear to converge to A099773.

A bisected even version is A116598 (counts partitions by number of 1's).

Counting all parts (not just > 1) gives A331416, shifted A331385.

A000041 counts integer partitions, strict A000009 (also into odds).

A113685 counts partitions by sum of odd parts, rank statistic A366528.

A330953 counts partitions with Heinz number divisible by sum of primes.

A331381 counts partitions with (product)|(sum of primes), equality A331383.

Cf. A000837, A014689, A113686, A239261, A331417, A331418, A365067, A366842.

Sequence in context: A300060 A300056 A321758 * A357525 A016102 A083905

Adjacent sequences: A366848 A366849 A366850 * A366852 A366853 A366854

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 01 2023

Status

approved