A352486 Heinz numbers of non-self-conjugate integer partitions.

3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Author

Gus Wiseman, Mar 20 2022

Keywords

• nonn

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions. The sequence lists all Heinz numbers of partitions whose Heinz number is different from that of their conjugate.

Examples

			The terms together with their prime indices begin:
   3: (2)
   4: (1,1)
   5: (3)
   7: (4)
   8: (1,1,1)
  10: (3,1)
  11: (5)
  12: (2,1,1)
  13: (6)
  14: (4,1)
  15: (3,2)
  16: (1,1,1,1)
  17: (7)
  18: (2,2,1)
For example, the self-conjugate partition (4,3,3,1) has Heinz number 350, so 350 is not in the sequence.
		

Crossrefs

The complement is A088902, counted by A000700.

These partitions are counted by A330644.

These are the positions of nonzero terms in A352491.

A000041 counts integer partitions, strict A000009.

A098825 counts permutations by unfixed points.

A238349 counts compositions by fixed points, rank statistic A352512.

A325039 counts partitions w/ same product as conjugate, ranked by A325040.

A352523 counts compositions by unfixed points, rank statistic A352513.

Heinz number (rank) and partition:

- A003963 = product of partition, conjugate A329382

- A008480 = number of permutations of partition, conjugate A321648.

- A056239 = sum of partition

- A122111 = rank of conjugate partition

- A296150 = parts of partition, reverse A112798, conjugate A321649

- A352487 = less than conjugate, counted by A000701

- A352488 = greater than or equal to conjugate, counted by A046682

- A352489 = less than or equal to conjugate, counted by A046682

- A352490 = greater than conjugate, counted by A000701

Cf. A000720, A026424, A120383, A175508, A195017, A238745, A301987, A304360, A316524, A324846, A350841.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y0]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#!=Times@@Prime/@conj[primeMS[#]]&]

Formula

a(n) != A122111(a(n)).