A209402 Number of partitions of n into distinct primes except the prime factors of n.
1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 4, 3, 2, 3, 1, 2, 6, 3, 8, 4, 2, 5, 3, 5, 10, 4, 3, 5, 13, 4, 14, 7, 4, 7, 18, 9, 12, 9, 6, 9, 25, 11, 10, 10, 9, 14, 34, 10, 38, 16, 11, 24, 14, 13, 49, 20, 18, 12, 60, 25, 66, 31, 17, 28
Offset: 0
Examples
a(5) = 1: [2,3]. a(7) = 1: [2,5]. a(16) = 2: [3,13], [5,11]. a(23) = 4: [3,7,13], [2,3,5,13], [5,7,11], [2,3,7,11]. a(24) = 3: [5,19], [7,17], [11,13]. a(29) = 6: [3,7,19], [2,3,5,19], [5,7,17], [2,3,7,17], [5,11,13], [2,3,11,13]. a(34) = 5: [3,31], [5,29], [11,23], [3,5,7,19], [3,7,11,13].
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..5000
Programs
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Maple
with(numtheory): a:= proc(n) option remember; local b, l, f; b:= proc(h, j) option remember; `if`(h=0, 1, `if`(j<1, 0, `if`(l[j]>h, 0, b(h-l[j], j-1)) +b(h, j-1))) end; forget(b); f:= factorset(n); l:= sort([({ithprime(i)$i=1..pi(n)} minus f)[]]); b(n, nops(l)) end: seq(a(n), n=0..300); -
Mathematica
a[n_] := a[n] = Module[{b, l, f}, b[h_, j_] := b[h, j] = If[h == 0, 1, If[j < 1, 0, If[l[[j]] > h, 0, b[h - l[[j]], j-1]] + b[h, j-1]]]; f = FactorInteger[n][[All, 1]]; l = Sort[Union[Array[Prime, PrimePi[n]] ~Complement~ f]]; b[n, Length[l]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 27 2017, translated from Maple *)
Formula
a(p) = A000586(p)-1 for any prime p.
Comments