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A116864 Array of product of parts of the partitions of n with only prime parts.

%I A116864 #12 Aug 29 2019 16:20:16
%S A116864 0,2,0,3,0,0,0,0,4,0,0,5,0,6,0,0,0,0,0,0,0,9,0,0,8,0,0,0,0,7,0,10,0,0,
%T A116864 0,0,12,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,18,0,0,0,0,16,0,0,0,0,0,0,0,
%U A116864 0,0,14,0,0,0,0,0,0,20,0,27
%N A116864 Array of product of parts of the partitions of n with only prime parts.
%C A116864 The inverse of sequence A001414 (sopfr(n)=sum of prime factors of n). See the examples and the W. Lang link.
%C A116864 The row length sequence of this array is p(n)=A000041(n) (number of partitions).
%C A116864 The partitions of n are ordered according to Abramowitz-Stegun (A-St), pp. 831-2.
%C A116864 Row n gives the values k for which A001414(k)=n>=2. E.g. n=10 appears 5 times in A001414, namely for the k values 21, 25, 30, 36 and 32.
%H A116864 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A116864 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.
%H A116864 W. Lang: <a href="/A116864/a116864.txt">First 10 rows.</a>
%F A116864 a(n,k)=product(part(i),i=1..m(n,k)) if the k-th partition of n in the A-St order has only prime parts. Here m(n,k) is the number of parts of this partition. Otherwise a(n,k)=0. See A000040 for the prime numbers.
%e A116864 [0];
%e A116864 [2, 0];
%e A116864 [3, 0, 0];
%e A116864 [0, 0, 4, 0, 0];
%e A116864 [5, 0, 6, 0, 0, 0, 0];
%e A116864 ...
%e A116864 a(4,3)=4 because the third partition of 4 is, in A-St order, (2,2)
%e A116864 with product 4. There is only this partition of 4 with only prime parts.
%e A116864 Row n=5 shows: n=5 appears twice in A001414(k), namely for k= 5 and
%e A116864 6. This is related to the two partitions (5) and (3,2) with only prime parts.
%Y A116864 Row sums give A002098(n), n>=1.
%Y A116864 Row sums (with nonzero numbers replaced by 1) give A000607(n), n>=1. See the array A116865.
%K A116864 nonn,easy,tabf
%O A116864 1,2
%A A116864 _Wolfdieter Lang_, Mar 24 2006