A182629 Total number of largest parts in all partitions of n that contain at least two distinct parts.

0, 0, 0, 1, 2, 6, 8, 17, 23, 36, 51, 75, 95, 138, 181, 236, 310, 407, 516, 667, 840, 1062, 1344, 1678, 2080, 2589, 3212, 3942, 4851, 5937, 7246, 8824, 10724, 12971, 15705, 18895, 22749, 27296, 32734, 39083, 46668, 55553, 66086, 78389, 92937, 109857, 129850

Offset: 0

Author

Omar E. Pol, Jul 14 2011

Keywords

• nonn

Comments

a(n) is also the sum of smallest parts of all partitions of n minus the sum of divisors of n, for n >= 1.

Examples

			For n = 6 the partitions of 6 are
6 ....................... all parts are equal.
5 + 1 ................... contains only one largest part.
4 + 2 ................... contains only one largest part.
4 + 1 + 1 ............... contains only one largest part.
3 + 3 ................... all parts are equal.
3 + 2 + 1 ............... contains only one largest part.
3 + 1 + 1 + 1 ........... contains only one largest part.
2 + 2 + 2 ............... all parts are equal.
2 + 2 + 1 + 1 ........... contains two largest parts.
2 + 1 + 1 + 1 + 1 ....... contains only one largest part.
1 + 1 + 1 + 1 + 1 + 1 ... all parts are equal.
There are 8 largest parts, so a(6) = 8.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000203, A006128, A046746, A144300.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=i, n, 0)+
      `if`(i<1, 0, b(n, i-1) +`if`(n b(n, n) -numtheory[sigma](n):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 17 2013

Mathematica

b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i - 1] + If[n < i, 0, b[n - i, i]]]; a[n_] := b[n, n] - DivisorSigma[1, n]; a[0] = 0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 06 2017, after Alois P. Heinz *)

Formula

a(n) = A046746(n) - A000203(n), for n >= 1. - Omar E. Pol, Jul 15 2011

Extensions

More terms a(13)-a(46) from David Scambler, Jul 15 2011