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%I #9 May 08 2017 00:33:19
%S 1,0,1,0,1,1,1,1,1,1,3,1,3,1,3,3,3,4,3,4,6,4,7,4,7,7,7,9,8,9,12,9,14,
%T 10,15,14,15,17,17,18,22,19,25,21,27,27,28,31,31,33,38,36,42,39,45,47,
%U 49,52,55,55,64,61,70,67,74,77,81,84,91,89,102,98,110,109,116,123,126,133,141,141,156,153,168,169,178,188,193
%N Expansion of Product_{j>=1} 1/(1 - x^(Sum_{i=1..j} prime(i))).
%C Number of partitions of n into nonzero partial sums of primes (A007504).
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePartition.html">Prime Partition</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F G.f.: Product_{j>=1} 1/(1 - x^(Sum_{i=1..j} prime(i))).
%e a(10) = 3 because we have [10], [5, 5] and [2, 2, 2, 2, 2], where 2 = prime(1), 5 = prime(1) + prime(2), 10 = prime(1) + prime(2) + prime(3).
%t nmax = 86; CoefficientList[Series[Product[1/(1 - x^Sum[Prime[i], {i, 1, j}]), {j, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000586, A000607, A007504, A281274.
%K nonn
%O 0,11
%A _Ilya Gutkovskiy_, Jan 18 2017
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