A349151 Heinz numbers of integer partitions with alternating sum <= 1.

1, 2, 4, 6, 8, 9, 15, 16, 18, 24, 25, 32, 35, 36, 49, 50, 54, 60, 64, 72, 77, 81, 96, 98, 100, 121, 128, 135, 140, 143, 144, 150, 162, 169, 196, 200, 216, 221, 225, 240, 242, 256, 288, 289, 294, 308, 315, 323, 324, 338, 361, 375, 384, 392, 400, 437, 441, 450

Offset: 1

Author

Gus Wiseman, Nov 10 2021

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 alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so these are also Heinz numbers of partitions with at most one odd conjugate part.

Examples

			The terms and their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   25: {3,3}
   32: {1,1,1,1,1}
   35: {3,4}
   36: {1,1,2,2}
   49: {4,4}
		

Crossrefs

The case of alternating sum 0 is A000290.

These partitions are counted by A100824.

These are the positions of 0's and 1's in A344616.

The case of alternating sum 1 is A345958.

The conjugate partitions are ranked by A349150.

A000041 counts integer partitions.

A056239 adds up prime indices, row sums of A112798.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A106529 ranks balanced partitions, counted by A047993.

A122111 is a representation of partition conjugation.

A257991 counts odd prime indices.

A316524 gives the alternating sum of prime indices.

A344610 counts partitions by sum and positive reverse-alternating sum.

A349157 ranks partitions with as many even parts as odd conjugate parts.

Cf. A000070, A000700, A001222, A027187, A027193, A215366, A277103, A277579, A326841, A349149, A349158.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[100],ats[Reverse[primeMS[#]]]<=1&]

Formula

Equals A000290 \/ A345958 decapitated.