A067953 Number of ways to sum numbers from 1 to n to the n-th prime.
0, 1, 1, 2, 2, 4, 7, 13, 23, 39, 69, 122, 211, 339, 564, 1001, 1764, 2630, 4565, 7192, 10151, 17202, 26152, 43543, 79126, 117496, 156229, 227302, 295011, 422040, 1004905, 1423445, 2210752, 2796140, 5225780, 6546101, 9921635, 14947534
Offset: 1
Examples
a(6)=4, as there are 4 decompositions for A000040(6)=13: 6+5+2 = 6+4+3 = 6+4+2+1 = 5+4+3+1 = 13.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Programs
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Haskell
a067953 n = p [1..n] $ a000040 n where p _ 0 = 1 p [] _ = 0 p (k:ks) m | m < k = 0 | otherwise = p ks (m - k) + p ks m -- Reinhard Zumkeller, Nov 22 2011
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Mathematica
(* This program is not convenient for a large number of terms *) a[n_] := Count[ IntegerPartitions[ Prime[n], {2, Floor[n/2 + 2]}, Range[n] ], nn_ /; Length[nn] == Length[nn // Union] ]; Table[Print[n, " ", an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Jun 25 2013 *) -
PARI
a(n)=my(p=prime(n),x='x);polcoeff(prod(k=1,n,1+x^k, O(x^(p+1))+1), p) \\ Charles R Greathouse IV, Jun 25 2013
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PARI
first(n)=my(v=vector(n),x='x,P=O(x^(prime(n)+1))+1,i);forprime(p=2,prime(n),P*=1+x^i++;v[i]=polcoeff(P,p));v \\ Charles R Greathouse IV, Jun 25 2013
Formula
Extensions
a(36)-a(38) from Donovan Johnson, Aug 23 2010