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
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
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