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

0,8

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).

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)

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A000041, A002865, A032741, A144300, A135010, A138121, A167929, A167930, A167932, A167934.

Sequence in context: A032720 A289117 A355697 * A090867 A152950 A005626

Adjacent sequences: A167925 A167926 A167927 * A167929 A167930 A167931

Keyword

nonn

Author

Omar E. Pol, Nov 17 2009

Extensions

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

More terms from Alois P. Heinz, May 24 2013

Status

approved