A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 3, 3, 5, 1, 7, 1, 8, 4, 6, 1, 15, 2, 9, 5, 14, 1, 22, 1, 20, 7, 18, 4, 36, 1, 26, 10, 40, 1, 51, 1, 48, 18, 49, 1, 86, 3, 73, 19, 86, 1, 117, 7, 120, 27, 120, 1, 196, 1, 160, 42, 201, 10, 259, 1, 258, 50, 292, 1, 407, 1, 357, 81, 431, 8, 548, 1, 577

Offset: 1

Author

Vladeta Jovovic, Oct 23 2004

Keywords

• easy
• nonn

Comments

If n > 0, we can assume this part is the smallest. - Gus Wiseman, Apr 18 2021

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

The non-strict version with 1's allowed is A083710.

The non-strict version is A083711.

The version with 1's allowed is A097986.

The Heinz numbers of these partitions are the odd terms of A339563.

The non-strict dual is A339619.

The strict complement is counted by A341450.

A000005 counts divisors.

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.

A018818 counts partitions into divisors (strict: A033630).

A167865 counts strict chains of divisors > 1 summing to n.

Cf. A130689, A130714, A200745, A264401, A343345, A343347, A343378, A343381.

Programs

Mathematica

Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 92}], {k, 2, 92}]], x], {2, 81}] (* Robert G. Wilson v, Nov 01 2004 *)
Table[If[n==0,0,Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&]]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

Formula

a(n) = Sum_{d|n, dA025147(d-1).

G.f.: Sum_{k>=2} (x^k*Product_{i>=2}(1 + x^(k*i))).

Extensions

More terms from Robert G. Wilson v, Nov 01 2004

Name shortened by Gus Wiseman, Apr 23 2021