A220483 Total number of smallest parts that are also emergent parts in all partitions of n with at least one distinct part: a(n) = n + d(n) + p(n-1) + spt(n) - A000070(n) - sigma(n) - 1.

0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 8, 11, 19, 26, 34, 51, 67, 91, 118, 158, 200, 271, 331, 433, 538, 699, 849, 1089, 1323, 1674, 2030, 2542, 3066, 3813, 4567, 5640, 6760, 8272, 9871, 12002, 14290, 17287, 20515, 24675, 29214, 34981, 41282, 49216, 57957, 68798

Offset: 1

Author

Omar E. Pol, Jan 16 2013

Keywords

• nonn

Comments

For the definition of "emergent part" see A182699, A182709.

Crossrefs

Cf. A000005, A000041, A000070, A000203, A002865, A092269, A182699, A182709, A183152, A193827, A195820, A206437, A215513, A220479, A220489.

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}]];
a[n_] := n + DivisorSigma[0, n] + PartitionsP[n - 1] + b[n, n] -
  Total[PartitionsP[Range[0, n]]] - DivisorSigma[1, n] - 1;
Array[a, 50] (* Jean-François Alcover, Jun 05 2021, using Alois P. Heinz's code for A092269 *)

Formula

a(n) = n + A000005(n) + A000041(n-1) + A092269(n) - A000070(n) - A000203(n) - 1.

Extensions

a(49) corrected by Jean-François Alcover, Jun 05 2021