A167928 Number of partitions of n that do not contain 1 as a part and whose parts are not the same divisor of n.

1, 0, 0, 0, 0, 1, 1, 3, 4, 6, 9, 13, 16, 23, 31, 38, 51, 65, 83, 104, 132, 162, 207, 252, 313, 381, 475, 571, 703, 846, 1032, 1237, 1502, 1791, 2164, 2570, 3086, 3659, 4375, 5167, 6146, 7244, 8584, 10086, 11909, 13954, 16421, 19195, 22510, 26250, 30696, 35714

Offset: 0

Author

Omar E. Pol, Nov 17 2009

Keywords

• nonn

Comments

Note that these partitions are located in the head of the last section of the set of partitions of n (see the shell model of partitions, here).

Examples

			The partitions of 6 are:
6 ....................... All parts are the same divisor of n.
5 + 1 ................... Contains 1 as a part.
4 + 2 ................... All parts are not the same divisor of n. <------(1)
4 + 1 + 1 ............... Contains 1 as a part.
3 + 3 ................... All parts are the same divisor of n.
3 + 2 + 1 ............... Contains 1 as a part.
3 + 1 + 1 + 1 ........... Contains 1 as a part.
2 + 2 + 2 ............... All parts are the same divisor of n.
2 + 2 + 1 + 1 ........... Contains 1 as a part.
2 + 1 + 1 + 1 + 1 ....... Contains 1 as a part.
1 + 1 + 1 + 1 + 1 + 1 ... Contains 1 as a part.
Then a(6) = 1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• 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. A000041, A002865, A032741, A144300, A135010, A138121, A167929, A167930, A167932, A167934.

Programs

Maple

b:= proc(n, i, t) option remember;
      `if`(n=0, `if`(t<>1, 1, 0), `if`(i<2, 0,
      add(b(n-i*j, i-1, `if`(j=0, t, max(0, t-1))), j=0..n/i)))
    end:
a:= n-> b(n, n, 2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 24 2013

Mathematica

Prepend[Array[ n \[Function] Length@Select[IntegerPartitions[n, All, Range[2, n - 1]], Length[Union[ # ]] > 1 &], 40], 1] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t != 1, 1, 0], If[i < 2, 0, Sum[b[n - i*j, i - 1, If[j == 0, t, Max[0, t - 1]]], {j, 0, n/i}]]]; a[n_] := b[n, n, 2]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

Formula

a(n) = A002865(n) - A032741(n).

Extensions

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

More terms from Alois P. Heinz, May 24 2013