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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
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).
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 *)
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