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A235945
Number of partitions of n containing at least one prime.
10
0, 0, 1, 2, 3, 5, 8, 12, 17, 24, 34, 48, 65, 88, 118, 157, 205, 269, 348, 450, 575, 734, 929, 1176, 1473, 1845, 2297, 2856, 3527, 4355, 5346, 6558, 8004, 9759, 11848, 14374, 17363, 20958, 25210, 30292, 36278, 43412, 51792, 61733, 73383, 87146, 103239, 122194
OFFSET
0,4
LINKS
FORMULA
a(n) = A000041(n) - A002095(n).
Product_{k>0} 1/(1-x^k) - Product_{k>0} (1-x^prime(k))/(1-x^k). - Alois P. Heinz, Jan 18 2014
EXAMPLE
a(5) = 5 because 5 partitions of 5 contain at least one prime: [5], [3,2], [3,1,1], [2,2,1], [2,1,1,1].
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i>n or isprime(i), 0, b(n-i, i))))
end:
a:= n-> combinat[numbpart](n) -b(n, n):
seq(a(n), n=0..50); # Alois P. Heinz, Jan 18 2014
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n || PrimeQ[i], 0, b[n-i, i]]]]; a[n_] := PartitionsP[n]-b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 28 2014, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A280276 A233969 A240202 * A129504 A241553 A241549
KEYWORD
nonn
AUTHOR
J. Stauduhar, Jan 17 2014
STATUS
approved