A339619 - OEIS

A339619

Number of integer partitions of n with no 1's and a part divisible by all the other parts.

1, 0, 1, 1, 2, 1, 4, 1, 5, 3, 7, 2, 13, 2, 13, 9, 17, 6, 27, 7, 33, 19, 35, 16, 58, 22, 58, 39, 75, 37, 108, 44, 117, 75, 132, 88, 190, 94, 199, 147, 250, 153, 322, 180, 363, 271, 405, 286, 544, 339, 601, 458, 699, 503, 868, 608, 990, 777, 1113, 865, 1422, 993

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OFFSET

0,5

COMMENTS

Alternative name: Number of integer partitions of n with no 1's that are empty or have greatest part divisible by all the other parts.

LINKS

Table of n, a(n) for n=0..61.

EXAMPLE

The a(6) = 4 through a(16) = 17 partitions (A..G = 10..16):

6 7 8 9 A B C D E F G

33 44 63 55 632 66 6322 77 A5 88

42 62 333 82 84 C2 C3 C4

222 422 442 93 662 555 E2

2222 622 A2 842 663 844

4222 444 A22 933 C22

22222 633 4442 6333 4444

822 6332 33333 6622

3333 8222 63222 8422

4422 44222 A222

6222 62222 44422

42222 422222 63322

222222 2222222 82222

442222

622222

4222222

22222222

MATHEMATICA

Table[If[n==0, 1, Length[Select[IntegerPartitions[n], FreeQ[#, 1]&&Or@@And@@IntegerQ/@(Max@@#/#)&]]], {n, 0, 30}]

CROSSREFS

The dual version is A083711.

The version with 1's allowed is A130689.

The strict case is A339660.

The Heinz numbers of these partitions are the odd complement of A343337.

The strict case with 1's allowed is A343347.

A000009 counts strict partitions.

A000041 counts partitions.

A000070 counts partitions with a selected part.

A006128 counts partitions with a selected position.

A015723 counts strict partitions with a selected part.

Cf. A066186, A083710, A083711, A097986, A130714, A338470, A343341, A343342, A343346, A343377, A343382.

Sequence in context: A339243 A214579 A083711 * A018783 A200976 A328187

Adjacent sequences: A339616 A339617 A339618 * A339620 A339621 A339622

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 18 2021

STATUS

approved