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
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
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
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Jan 16 2013
STATUS
approved