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A325507 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all integer partitions of n.

%I A325507 #8 May 07 2019 23:14:32
%S A325507 1,2,6,28,340,3108,106932,2732340,236790060,19703562780,3419598096420,
%T A325507 674127752953380,264134168649181380,95825592671995399620,
%U A325507 67662122741507082338220,50556978553034312461203420,69259146896604886347745839660,104191622563656655781003976625020
%N A325507 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all integer partitions of n.
%C A325507 Also the Heinz number of row n of A066633.
%C A325507 The Heinz number of an integer partition or sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A325507 <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>
%F A325507 a(n) = Product_{i = 1..n} prime(A066633(n,i)).
%F A325507 a(n) = A181819(A003963(A325500(n))).
%F A325507 a(n) = A181819(A325501(n)).
%F A325507 A001222(a(n)) = n.
%F A325507 A056239(a(n)) = A006128(n).
%F A325507 For n > 0, A181819(a(n)) = A087009(n + 1).
%e A325507 The integer partitions of 4 are {(4), (3,1), (2,2), (2,1,1), (1,1,1,1)}, with multiset union {1,1,1,1,1,1,1,2,2,2,3,4}, with multiplicities (7,3,1,1), so a(4) = prime(7)*prime(3)*prime(1)*prime(1) = 340.
%e A325507 The sequence of terms together with their prime indices begins:
%e A325507                         1: {}
%e A325507                         2: {1}
%e A325507                         6: {1,2}
%e A325507                        28: {1,1,4}
%e A325507                       340: {1,1,3,7}
%e A325507                      3108: {1,1,2,4,12}
%e A325507                    106932: {1,1,2,4,8,19}
%e A325507                   2732340: {1,1,2,3,6,11,30}
%e A325507                 236790060: {1,1,2,3,6,9,19,45}
%e A325507               19703562780: {1,1,2,3,5,8,15,26,67}
%e A325507             3419598096420: {1,1,2,3,5,8,13,21,41,97}
%e A325507           674127752953380: {1,1,2,3,5,7,12,18,31,56,139}
%e A325507        264134168649181380: {1,1,2,3,5,7,12,17,28,45,83,195}
%e A325507      95825592671995399620: {1,1,2,3,5,7,11,16,25,38,63,112,272}
%e A325507   67662122741507082338220: {1,1,2,3,5,7,11,16,24,35,55,87,160,373}
%t A325507 Table[Times@@Prime/@Length/@Split[Sort[Join@@IntegerPartitions[n]]],{n,0,15}]
%Y A325507 Cf. A001222, A003963, A006128, A007870, A056239, A066633, A087009, A112798, A215366, A302246, A325500, A325501, A325513.
%K A325507 nonn
%O A325507 0,2
%A A325507 _Gus Wiseman_, May 07 2019