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

0,2

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).

Links

Table of n, a(n) for n=0..17.

Index entries for sequences related to Heinz numbers

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}]

Crossrefs

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

Sequence in context: A355768 A277480 A108800 * A306793 A326359 A136639

Adjacent sequences: A325504 A325505 A325506 * A325508 A325509 A325510

Keyword

nonn

Author

Gus Wiseman, May 07 2019

Status

approved