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

1,6

Comments

For the definition of emergent parts see A182699.

Links

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

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

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 *)

Crossrefs

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

Sequence in context: A339419 A213952 A205873 * A146913 A146252 A181641

Adjacent sequences: A220476 A220477 A220478 * A220480 A220481 A220482

Keyword

nonn

Author

Omar E. Pol, Jan 12 2013

Extensions

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

Status

approved