A220483 - OEIS

%I #17 Jun 05 2021 16:33:51

%S 0,0,0,0,0,0,1,1,3,5,8,11,19,26,34,51,67,91,118,158,200,271,331,433,

%T 538,699,849,1089,1323,1674,2030,2542,3066,3813,4567,5640,6760,8272,

%U 9871,12002,14290,17287,20515,24675,29214,34981,41282,49216,57957,68798

%N 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.

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

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

%t 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}]];

%t a[n_] := n + DivisorSigma[0, n] + PartitionsP[n - 1] + b[n, n] -

%t   Total[PartitionsP[Range[0, n]]] - DivisorSigma[1, n] - 1;

%t Array[a, 50] (* _Jean-François Alcover_, Jun 05 2021, using _Alois P. Heinz_'s code for A092269 *)

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

%K nonn

%O 1,9

%A _Omar E. Pol_, Jan 16 2013

%E a(49) corrected by _Jean-François Alcover_, Jun 05 2021