cp's OEIS Frontend

A182629 Total number of largest parts in all partitions of n that contain at least two distinct parts.

%I A182629 #30 Feb 06 2017 08:11:29
%S A182629 0,0,0,1,2,6,8,17,23,36,51,75,95,138,181,236,310,407,516,667,840,1062,
%T A182629 1344,1678,2080,2589,3212,3942,4851,5937,7246,8824,10724,12971,15705,
%U A182629 18895,22749,27296,32734,39083,46668,55553,66086,78389,92937,109857,129850
%N A182629 Total number of largest parts in all partitions of n that contain at least two distinct parts.
%C A182629 a(n) is also the sum of smallest parts of all partitions of n minus the sum of divisors of n, for n >= 1.
%H A182629 Alois P. Heinz, <a href="/A182629/b182629.txt">Table of n, a(n) for n = 0..1000</a>
%F A182629 a(n) = A046746(n) - A000203(n), for n >= 1. - _Omar E. Pol_, Jul 15 2011
%e A182629 For n = 6 the partitions of 6 are
%e A182629 6 ....................... all parts are equal.
%e A182629 5 + 1 ................... contains only one largest part.
%e A182629 4 + 2 ................... contains only one largest part.
%e A182629 4 + 1 + 1 ............... contains only one largest part.
%e A182629 3 + 3 ................... all parts are equal.
%e A182629 3 + 2 + 1 ............... contains only one largest part.
%e A182629 3 + 1 + 1 + 1 ........... contains only one largest part.
%e A182629 2 + 2 + 2 ............... all parts are equal.
%e A182629 2 + 2 + 1 + 1 ........... contains two largest parts.
%e A182629 2 + 1 + 1 + 1 + 1 ....... contains only one largest part.
%e A182629 1 + 1 + 1 + 1 + 1 + 1 ... all parts are equal.
%e A182629 There are 8 largest parts, so a(6) = 8.
%p A182629 b:= proc(n, i) option remember; `if`(n=i, n, 0)+
%p A182629       `if`(i<1, 0, b(n, i-1) +`if`(n<i, 0, b(n-i, i)))
%p A182629     end:
%p A182629 a:= n-> b(n, n) -numtheory[sigma](n):
%p A182629 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jan 17 2013
%t A182629 b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i - 1] + If[n < i, 0, b[n - i, i]]]; a[n_] := b[n, n] - DivisorSigma[1, n]; a[0] = 0; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Feb 06 2017, after _Alois P. Heinz_ *)
%Y A182629 Cf. A000203, A006128, A046746, A144300.
%K A182629 nonn
%O A182629 0,5
%A A182629 _Omar E. Pol_, Jul 14 2011
%E A182629 More terms a(13)-a(46) from _David Scambler_, Jul 15 2011