A167934 a(n) = A000041(n) - A032741(n).

1, 1, 1, 2, 3, 6, 8, 14, 19, 28, 39, 55, 72, 100, 132, 173, 227, 296, 380, 489, 622, 789, 999, 1254, 1568, 1956, 2433, 3007, 3713, 4564, 5597, 6841, 8344, 10140, 12307, 14880, 17969, 21636, 26012, 31182, 37331, 44582, 53167, 63260, 75170

Offset: 0

Author

Omar E. Pol, Nov 16 2009

Keywords

• nonn

Comments

a(n) is also the number of partitions of n whose parts are not all equal, (including however the partition with a single part of size n). Note that the number of partitions of n whose parts are all equal gives the number of divisors of n, for n>0. (See also A144300.)

Examples

			The partitions of n = 6 are:
6 ....................... All parts are equal, but included .. (1).
5 + 1 ................... All parts are not equal ............ (2).
4 + 2 ................... All parts are not equal ............ (3).
4 + 1 + 1 ............... All parts are not equal ............ (4).
3 + 3 ................... All parts are equal, not included.
3 + 2 + 1 ............... All parts are not equal ............ (5).
3 + 1 + 1 + 1 ........... All parts are not equal ............ (6).
2 + 2 + 2 ............... All parts are equal, not included.
2 + 2 + 1 + 1 ........... All parts are not equal ............ (7).
2 + 1 + 1 + 1 + 1 ....... All parts are not equal ............ (8).
1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal, not included.
Then a(6) = 8.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)
• Omar E. Pol, Illustration of the shell model of partitions (2D view)
• Omar E. Pol, Illustration of the shell model of partitions (3D view)

Crossrefs

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A144300, A135010, A138121, A167930, A167932, A167935.

Programs

Maple

b:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then `if`(k=0, 1, 0)
    elif i=0 then 0
    else b(n, i-1, k)+
         b(n-i, i, `if`(k<0, i, `if`(k<>i, 0, k)))
      fi
    end:
a:= n-> 1 +b(n, n-1, -1):
seq(a(n), n=0..50);  #  Alois P. Heinz, Dec 01 2010

Mathematica

a[0] = 1; a[n_] := PartitionsP[n] - DivisorSigma[0, n] + 1; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 08 2016 *)

Formula

a(n) = A000041(n) - A032741(n).

Extensions

More terms from Alois P. Heinz, Dec 01 2010