A060034 Number of partitions of n such that all parts are neither relatively prime (cf. A000837) nor are they periodic with each part occurring the same number of times (cf. A024994).

0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 4, 0, 9, 3, 12, 0, 22, 0, 28, 9, 43, 0, 63, 3, 82, 19, 107, 0, 170, 0, 189, 43, 258, 12, 372, 0, 435, 82, 557, 0, 808, 0, 900, 162, 1150, 0, 1599, 9, 1836, 258, 2252, 0, 3111, 46, 3476, 435, 4308, 0, 5827, 0, 6501, 727, 7917, 85

Offset: 1

Author

Alford Arnold, Mar 16 2001

Keywords

• easy
• nonn

Examples

			a(15) = 3 because partitions 6+3+3+3, 6+6+3 and 9+3+3 satisfy the description and A000041(15) - (A000837(15) + A024994(15)) = 176 - (167 + 6) = 3.
		

Crossrefs

A000041, A000837, A024994 and A055892.

Programs

Mathematica

A000837[n_] := Sum[ MoebiusMu[n/d]*PartitionsP[d], {d, Divisors[n]}]; A024994[n_] := Sum[ PartitionsQ[k], {k, Divisors[n] // Most}]; a[n_] := PartitionsP[n] - (A000837[n] + A024994[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 03 2013 *)

Formula

a(n) = A000041(n) - ( A000837(n) + A024994(n))

Extensions

More terms from Naohiro Nomoto, Mar 01 2002