A300486 Number of relatively prime or monic partitions of n.

1, 2, 3, 4, 7, 8, 15, 18, 28, 35, 56, 64, 101, 120, 168, 210, 297, 348, 490, 583, 776, 946, 1255, 1482, 1952, 2335, 2981, 3581, 4565, 5387, 6842, 8119, 10086, 12013, 14863, 17527, 21637, 25525, 31083, 36695, 44583, 52256, 63261, 74171, 88932, 104303, 124754

Offset: 1

Author

Gus Wiseman, Apr 15 2018

Keywords

• nonn

Comments

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime).

Examples

			The a(6) = 8 relatively prime or monic partitions are (6), (51), (411), (321), (3111), (2211), (21111), (111111). Missing from this list are (42), (33), (222).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Crossrefs

Cf. A000837, A001383, A063834, A093637, A196545, A281113, A289501, A300383, A301462, A301467, A301480, A302094, A302698, A302915, A302916, A302917.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&]],{n,20}]

PARI

a(n)={(n > 1) + sumdiv(n, d, moebius(d)*numbpart(n/d))} \\ Andrew Howroyd, Aug 29 2018

Formula

a(n > 1) = 1 + A000837(n) = 1 + Sum_{d|n} mu(d) * A000041(n/d).