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

1, 2, 6, 28, 340, 3108, 106932, 2732340, 236790060, 19703562780, 3419598096420, 674127752953380, 264134168649181380, 95825592671995399620, 67662122741507082338220, 50556978553034312461203420, 69259146896604886347745839660, 104191622563656655781003976625020

Offset: 0

Author

Gus Wiseman, May 07 2019

Keywords

• nonn

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).

Examples

			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}
		

Links

• Index entries for sequences related to Heinz numbers

Crossrefs

Cf. A001222, A003963, A006128, A007870, A056239, A066633, A087009, A112798, A215366, A302246, A325500, A325501, A325513.

Programs

Mathematica

Table[Times@@Prime/@Length/@Split[Sort[Join@@IntegerPartitions[n]]],{n,0,15}]

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).