A215513 spt(n) - p(n): total number of smallest parts in all partitions of n minus the number of partitions of n.

0, 1, 2, 5, 7, 15, 20, 35, 50, 77, 105, 161, 214, 305, 413, 570, 751, 1022, 1330, 1772, 2295, 2996, 3837, 4970, 6305, 8050, 10155, 12844, 16065, 20169, 25055, 31197, 38549, 47650, 58540, 71960, 87916, 107424, 130655, 158830, 192260, 232642, 280406

Offset: 1

Author

Omar E. Pol, Jan 13 2013

Keywords

• nonn

Comments

Also total number of smallest parts that are not on the right border in all partitions of n.

Examples

			For n = 6 the partitions of 6 with the smallest parts that are not in the right border in brackets are
-----------------------------------------
.      Partitions of 6            Value
-----------------------------------------
.                       6           0
.                  [3]+ 3           1
.                   4 + 2           0
.              [2]+[2]+ 2           2
.                   5 + 1           0
.               3 + 2 + 1           0
.               4 +[1]+ 1           1
.           2 + 2 +[1]+ 1           1
.           3 +[1]+[1]+ 1           2
.       2 +[1]+[1]+[1]+ 1           3
.  [1]+[1]+[1]+[1]+[1]+ 1           5
--------------------------------------
.                           Total: 15
On the other hand the total number of smallest parts in all partitions of 6 is 26 and the number of partitions of 6 is 11, so a(6) = 26 - 11 = 15.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000041, A000070, A002865, A092269, A120452, A220479.

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_] := b[n, n] - PartitionsP[n];
Array[a, 50] (* Jean-François Alcover, Jun 05 2021, using Alois P. Heinz's code for A092269 *)

Formula

a(n) = A092269(n) - A000041(n).

a(n) = A000070(n-2) + A220479(n), n >= 2.

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*Pi*sqrt(2*n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 49*Pi^2/6912)/n). - Vaclav Kotesovec, Jul 31 2017