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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.
5
1, 2, 6, 28, 340, 3108, 106932, 2732340, 236790060, 19703562780, 3419598096420, 674127752953380, 264134168649181380, 95825592671995399620, 67662122741507082338220, 50556978553034312461203420, 69259146896604886347745839660, 104191622563656655781003976625020
OFFSET
0,2
COMMENTS
Also the Heinz number of row n of A066633.
The Heinz number of an integer partition or sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
FORMULA
a(n) = Product_{i = 1..n} prime(A066633(n,i)).
a(n) = A181819(A003963(A325500(n))).
a(n) = A181819(A325501(n)).
A001222(a(n)) = n.
A056239(a(n)) = A006128(n).
For n > 0, A181819(a(n)) = A087009(n + 1).
EXAMPLE
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.
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
6: {1,2}
28: {1,1,4}
340: {1,1,3,7}
3108: {1,1,2,4,12}
106932: {1,1,2,4,8,19}
2732340: {1,1,2,3,6,11,30}
236790060: {1,1,2,3,6,9,19,45}
19703562780: {1,1,2,3,5,8,15,26,67}
3419598096420: {1,1,2,3,5,8,13,21,41,97}
674127752953380: {1,1,2,3,5,7,12,18,31,56,139}
264134168649181380: {1,1,2,3,5,7,12,17,28,45,83,195}
95825592671995399620: {1,1,2,3,5,7,11,16,25,38,63,112,272}
67662122741507082338220: {1,1,2,3,5,7,11,16,24,35,55,87,160,373}
MATHEMATICA
Table[Times@@Prime/@Length/@Split[Sort[Join@@IntegerPartitions[n]]], {n, 0, 15}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 07 2019
STATUS
approved