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A219203
Number of partitions of n into 9 distinct primes.
5
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 14 2012
STATUS
approved