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

%I A215513 #24 Jun 05 2021 16:33:26
%S A215513 0,1,2,5,7,15,20,35,50,77,105,161,214,305,413,570,751,1022,1330,1772,
%T A215513 2295,2996,3837,4970,6305,8050,10155,12844,16065,20169,25055,31197,
%U A215513 38549,47650,58540,71960,87916,107424,130655,158830,192260,232642,280406
%N A215513 spt(n) - p(n): total number of smallest parts in all partitions of n minus the number of partitions of n.
%C A215513 Also total number of smallest parts that are not on the right border in all partitions of n.
%H A215513 Vaclav Kotesovec, <a href="/A215513/b215513.txt">Table of n, a(n) for n = 1..10000</a>
%F A215513 a(n) = A092269(n) - A000041(n).
%F A215513 a(n) = A000070(n-2) + A220479(n), n >= 2.
%F A215513 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
%e A215513 For n = 6 the partitions of 6 with the smallest parts that are not in the right border in brackets are
%e A215513 -----------------------------------------
%e A215513 .      Partitions of 6            Value
%e A215513 -----------------------------------------
%e A215513 .                       6           0
%e A215513 .                  [3]+ 3           1
%e A215513 .                   4 + 2           0
%e A215513 .              [2]+[2]+ 2           2
%e A215513 .                   5 + 1           0
%e A215513 .               3 + 2 + 1           0
%e A215513 .               4 +[1]+ 1           1
%e A215513 .           2 + 2 +[1]+ 1           1
%e A215513 .           3 +[1]+[1]+ 1           2
%e A215513 .       2 +[1]+[1]+[1]+ 1           3
%e A215513 .  [1]+[1]+[1]+[1]+[1]+ 1           5
%e A215513 --------------------------------------
%e A215513 .                           Total: 15
%e A215513 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.
%t A215513 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 A215513 a[n_] := b[n, n] - PartitionsP[n];
%t A215513 Array[a, 50] (* _Jean-François Alcover_, Jun 05 2021, using _Alois P. Heinz_'s code for A092269 *)
%Y A215513 Cf. A000041, A000070, A002865, A092269, A120452, A220479.
%K A215513 nonn
%O A215513 1,3
%A A215513 _Omar E. Pol_, Jan 13 2013