cp's OEIS Frontend

A067953 Number of ways to sum numbers from 1 to n to the n-th prime.

%I A067953 #14 Jun 25 2013 09:52:38
%S A067953 0,1,1,2,2,4,7,13,23,39,69,122,211,339,564,1001,1764,2630,4565,7192,
%T A067953 10151,17202,26152,43543,79126,117496,156229,227302,295011,422040,
%U A067953 1004905,1423445,2210752,2796140,5225780,6546101,9921635,14947534
%N A067953 Number of ways to sum numbers from 1 to n to the n-th prime.
%H A067953 Charles R Greathouse IV, <a href="/A067953/b067953.txt">Table of n, a(n) for n = 1..1000</a>
%F A067953 a(n) = b(0, n), where b(m, n) satisfies b(m, n) = 1 + sum{b(i, j): m<i<j<n & i+j=prime(n)}.
%F A067953 a(n) < A000009(A000040(n)).
%e A067953 a(6)=4, as there are 4 decompositions for A000040(6)=13:
%e A067953 6+5+2 = 6+4+3 = 6+4+2+1 = 5+4+3+1 = 13.
%t A067953 (* 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 *)
%o A067953 (Haskell)
%o A067953 a067953 n = p [1..n] $ a000040 n where
%o A067953    p _  0 = 1
%o A067953    p [] _ = 0
%o A067953    p (k:ks) m | m < k = 0 | otherwise = p ks (m - k) + p ks m
%o A067953 -- _Reinhard Zumkeller_, Nov 22 2011
%o A067953 (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
%o A067953 (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
%K A067953 nonn,nice
%O A067953 1,4
%A A067953 _Reinhard Zumkeller_, Mar 06 2002
%E A067953 a(36)-a(38) from _Donovan Johnson_, Aug 23 2010