A220479 Total number of smallest parts that are also emergent parts in all partitions of n.

0, 0, 0, 1, 0, 3, 1, 5, 5, 10, 8, 22, 19, 33, 40, 62, 67, 107, 118, 175, 208, 282, 331, 462, 542, 712, 859, 1112, 1323, 1709, 2030, 2568, 3078, 3830, 4577, 5687, 6760, 8291, 9885, 12045, 14290, 17334, 20515, 24710, 29242, 35004, 41282, 49283, 57963, 68836

Offset: 1

Author

Omar E. Pol, Jan 12 2013

Keywords

• nonn

Comments

For the definition of emergent parts see A182699.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000041, A000070, A002865, A006128, A092269, A120452, A182699, A182709, A195820, A206437, A215513.

Programs

Mathematica

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r==0, q, 0] + Sum[b[n-i*j, i-1], {j, 0, n/i}]];
c[n_] := b[n, n];
d[n_] := Total[PartitionsP[Range[0, n-3]]] + PartitionsP[n-1];
a[n_] := c[n] - d[n+1];
Array[a, 50] (* Jean-François Alcover, Jun 05 2021, using Alois P. Heinz's code for A092269 *)

Formula

a(n) = A092269(n) - A000070(n-1) - A002865(n) = A092269(n) - A120452(n+1) = A195820(n) - A002865(n).

a(n) = A092269(n) - A000041(n) - A000070(n-2), n >= 2.

a(n) = A215513(n) - A000070(n-2), n >= 2.

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jul 31 2017

Extensions

a(43) corrected by Vaclav Kotesovec, Jul 31 2017