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A122697 Number of indecomposable partitions of n. 0
0, 2, 3, 2, 7, 5, 15, 14, 24, 28, 56, 52, 101, 105, 155, 189, 297, 310, 490, 536, 747, 890, 1255, 1380, 1930, 2234, 2928, 3433, 4565, 5133, 6842, 7881, 9975, 11716, 14778, 17006, 21637, 25035, 30882, 35972, 44583, 51200, 63261, 73115, 88459, 103048 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A partition is indecomposable if it is not [1] and cannot be represented as the product of two smaller partitions, where the product of two partitions is the multiset of all products of parts from the two multiplicands. Another way to define the product of partitions is to regard the partition as a finite sequence b(k) being the number of parts of size k; then the Dirichlet g.f. of b * c is the product of the Dirichlet g.f.s of b and c.

LINKS

Table of n, a(n) for n=1..46.

FORMULA

The (formal) Dirichlet generating function for A000041 is Product_{n>1} 1/(1-n^{-s})^a(n). (Formal because this g.f. does not converge for any value of s.)

EXAMPLE

The product of [2,2,1] * [2,1,1] is the partition with parts:

4 4 2

2 2 1

2 2 1

which is [4^2,2^5,1^2]. In terms of Dirichlet g.f.s, this is (2*2^s + 1^s) * (2^s + 2*1^s) = (2*4^s + 5*2^s + 2*1^s).

Of the partitions of 6, [6] = [3] * [2], [4,2] = [2] * [2,1], [3^2] = [3] * [1^2], [2^3] = [2] * [1^3], [2^2,1^2] = [2,1] * [1^2] and [1^6] = [1^3] * [1^2]. This leaves [5,1], [4,1^2], [3,2,1], [3,1^3] and [2,1^4] as the 5 indecomposable partitions of 6.

CROSSREFS

Cf. A000041, A090751.

Sequence in context: A158747 A260724 A256581 * A129022 A210564 A208930

Adjacent sequences:  A122694 A122695 A122696 * A122698 A122699 A122700

KEYWORD

nonn

AUTHOR

Franklin T. Adams-Watters, Sep 22 2006

STATUS

approved

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Last modified June 4 05:19 EDT 2020. Contains 334815 sequences. (Running on oeis4.)