A219203 Number of partitions of n into 9 distinct primes.

1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 5, 0, 4, 0, 2, 0, 9, 0, 7, 1, 6, 1, 13, 0, 10, 0, 12, 2, 20, 0, 19, 2, 20, 3, 31, 1, 30, 4, 28, 5, 49, 3, 45, 7, 43, 9, 69, 7, 63, 10, 66, 16, 97, 9, 91, 18, 96, 25, 130, 16, 131, 30, 134, 35, 177, 25, 182

Offset

100,13

Links

Alois P. Heinz, Table of n, a(n) for n = 100..10000

Formula

G.f.: Sum_{0<i_1<i_2<...<i_9} x^(Sum_{j=1..9} prime(i_j)).

a(n) = [x^n*y^9] Product_{i>=1} (1+x^prime(i)*y).

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0$9], `if`(i<1, [0$10],
       zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$9],
       b(n-ithprime(i), i-1)[1..9])[]], 0)))
    end:
a:= n-> b(n, numtheory[pi](n))[10]:
seq(a(n), n=100..180);

Mathematica

k = 9; 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, 100, 180}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n, {9}], AllTrue[#, PrimeQ]&&Length[Union[#]] == 9&]], {n, 100, 180}] (* Harvey P. Dale, Mar 09 2023 *)

Crossrefs

Column k=9 of A219180.

Sequence in context: A339012 A357984 A328766 * A341981 A356915 A226786

Adjacent sequences: A219200 A219201 A219202 * A219204 A219205 A219206

Keyword

nonn

Author

Alois P. Heinz, Nov 14 2012

Status

approved