A219200 Number of partitions of n into 6 distinct primes.
1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 4, 0, 2, 1, 5, 1, 6, 0, 5, 2, 6, 1, 10, 1, 9, 4, 11, 3, 15, 3, 14, 6, 16, 6, 22, 5, 20, 10, 25, 11, 29, 9, 29, 16, 34, 17, 39, 15, 39, 25, 45, 24, 50, 25, 53, 35, 57, 34, 66, 36, 68, 48, 75, 50, 83, 52, 88, 65, 92, 69, 104
Offset: 41
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 41..10000
Crossrefs
Column k=6 of A219180.
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0, [1,0$6], `if`(i<1, [0$7], zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$6], b(n-ithprime(i), i-1)[1..6])[]], 0))) end: a:= n-> b(n, numtheory[pi](n))[7]: seq(a(n), n=41..120); -
Mathematica
k = 6; b[n_, i_] := b[n, i] = If[n == 0, Join[{1}, Array[0&, k]], If[i<1, Array[0&, k+1], Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, Array[0&, k], Take[b[n-Prime[i], i-1], k]]]}]]]; a[n_] := b[n, PrimePi[n]][[k+1]]; Table[a[n], {n, 41, 120}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *) Table[Count[IntegerPartitions[n,{6}],?(AllTrue[#,PrimeQ]&&Length[Union[#]]==6&)],{n,41,120}] (* _Harvey P. Dale, Sep 17 2023 *)
Formula
G.f.: Sum_{0
a(n) = [x^n*y^6] Product_{i>=1} (1+x^prime(i)*y).