A347457 Heinz numbers of integer partitions with integer alternating product.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 71, 72, 73, 74, 75, 76, 78

Offset

1,2

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.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

Also numbers whose multiset of prime indices has integer reverse-alternating product.

Links

Table of n, a(n) for n=1..67.

Example

The prime indices of 525 are {2,3,3,4}, with reverse-alternating product 2, so 525 is in the sequence
The prime indices of 135 are {2,2,2,3}, with reverse-alternating product 3/2, so 135 is not in the sequence.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Select[Range[100], IntegerQ[altprod[Reverse[primeMS[#]]]]&]

Crossrefs

The reciprocal version is A028982.

Allowing any alternating product > 1 gives A028983, reverse A347465.

Factorizations of this type are counted by A347437.

These partitions are counted by A347446.

The reverse reciprocal version A347451.

The odd-length case is A347453.

The reverse version is A347454.

The complement is A347455.

A056239 adds up prime indices, row sums of A112798.

A316524 gives the alternating sum of prime indices (reverse: A344616).

A335433 lists numbers whose prime indices are separable, complement A335448.

A347461 counts possible alternating products of partitions, reverse A347462.

Cf. A001105, A001222, A028260, A119620, A119899, A316523, A344606, A344617, A346703, A346704, A347448, A347450.

Sequence in context: A247160 A023808 A331995 * A318991 A322901 A322911

Adjacent sequences: A347454 A347455 A347456 * A347458 A347459 A347460

Keyword

nonn

Author

Gus Wiseman, Sep 26 2021

Status

approved