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

41,7

Links

Alois P. Heinz, Table of n, a(n) for n = 41..10000

Formula

G.f.: Sum_{0<i_1<i_2<...<i_6} x^(Sum_{j=1..6} prime(i_j)).

a(n) = [x^n*y^6] Product_{i>=1} (1+x^prime(i)*y).

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 *)

Crossrefs

Column k=6 of A219180.

Sequence in context: A156667 A178090 A110914 * A341978 A193527 A127505

Adjacent sequences: A219197 A219198 A219199 * A219201 A219202 A219203

Keyword

nonn

Author

Alois P. Heinz, Nov 14 2012

Status

approved