A219201 Number of partitions of n into 7 distinct primes.
1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 4, 0, 3, 0, 3, 1, 6, 0, 6, 1, 5, 1, 10, 0, 11, 2, 9, 3, 16, 1, 17, 3, 15, 5, 25, 4, 24, 5, 25, 10, 35, 6, 34, 10, 36, 15, 48, 10, 50, 17, 52, 23, 65, 17, 69, 27, 70, 32, 89, 30, 93, 38, 93, 48, 116, 43, 121, 56, 125, 70, 148
Offset: 58
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 58..10000
Crossrefs
Column k=7 of A219180.
Programs
-
Maple
b:= proc(n, i) option remember; `if`(n=0, [1,0$7], `if`(i<1, [0$8], zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$7], b(n-ithprime(i), i-1)[1..7])[]], 0))) end: a:= n-> b(n, numtheory[pi](n))[8]: seq(a(n), n=58..140); -
Mathematica
k = 7; 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, 58, 140}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *) Table[Length@Select[IntegerPartitions[k,{7}, Prime@Range@100], #[[1]] > #[[2]] > #[[3]] > #[[4]] > #[[5]] > #[[6]] > #[[7]] &], {k, 58, 140}] (* Robert Price, Apr 25 2025 *)
Formula
G.f.: Sum_{0
a(n) = [x^n*y^7] Product_{i>=1} (1+x^prime(i)*y).