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A220477 Total number of parts in all partitions of n with at least one distinct part. 5
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified November 22 00:32 EST 2019. Contains 329383 sequences. (Running on oeis4.)