A220477 Total number of parts in all partitions of n with at least one distinct part.

0, 0, 2, 5, 14, 23, 46, 71, 115, 174, 263, 371, 542, 756, 1044, 1432, 1947, 2605, 3478, 4588, 6020, 7863, 10182, 13114, 16820, 21480, 27254, 34489, 43423, 54491, 68103, 84864, 105318, 130408, 160828, 197923, 242774, 297141, 362531, 441456, 536062, 649695

Offset

1,3

Comments

Also total number of parts in all partitions of n minus the sum of divisors of n. Also sum of largest parts of all partitions of n minus the sum of divisors of n.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

a(n) = A006128(n) - A000203(n).

G.f.: Q(0)/(1-x), where Q(k)= 1 - prod(n=1..k+1, (1-x^n))/( 1 - (x^(k+1)) - x*(1- (x^(k+1)))^2*(k+2)/( x*(1- (x^(k+1)))*(k+2) - (k+1)*(1 - (x^(k+2)))*prod(n=1..k+1, (1-x^n) )/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 16 2013

Example

For n = 6
-----------------------------------------------------
Partitions of 6            Value
-----------------------------------------------------
6 .......................... 0  (all parts are equal)
5 + 1 ...................... 2
4 + 2 ...................... 2
4 + 1 + 1 .................. 3
3 + 3 ...................... 0  (all parts are equal)
3 + 2 + 1 .................. 3
3 + 1 + 1 + 1 .............. 4
2 + 2 + 2 .................. 0  (all parts are equal)
2 + 2 + 1 + 1 .............. 4
2 + 1 + 1 + 1 + 1 .......... 5
1 + 1 + 1 + 1 + 1 + 1 ...... 0  (all parts are equal)
-----------------------------------------------------
The sum of the values is    23
On the other hand the total number of parts of the partitions of 6 is A006128(6) = 35 and the sum of divisor of 6 is 1 + 2 + 3 + 6 = sigma(6) = A000203(6) = 12 equals the total number of parts of the partitions of 6 into equal parts. So a(6) = 35 - 12 = 23.

Maple

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f, g:= b(n, i-1), `if`(i>n, [0$2], b(n-i, i));
         [f[1]+g[1], f[2]+g[2] +g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -numtheory[sigma](n):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 17 2013

Mathematica

a[n_] := Sum[DivisorSigma[0, k]*PartitionsP[n-k], {k, 1, n}] - DivisorSigma[1, n]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 22 2015 *)

Crossrefs

Cf. A000005, A000203, A000041, A006128, A066186, A182629, A182977, A182978.

Sequence in context: A131661 A321287 A076664 * A049939 A296302 A240401

Adjacent sequences: A220474 A220475 A220476 * A220478 A220479 A220480

Keyword

nonn,easy

Author

Omar E. Pol, Jan 16 2013

Status

approved