A347453 Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 42, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 114, 116, 117, 124, 125, 127, 128, 130

Offset

1,1

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 odd length and integer alternating product, where a prime index of n is a number m such that prime(m) divides n.

Links

Table of n, a(n) for n=1..63.

Example

The terms and their prime indices begin:
      2: {1}         29: {10}            61: {18}
      3: {2}         31: {11}            63: {2,2,4}
      5: {3}         32: {1,1,1,1,1}     67: {19}
      7: {4}         37: {12}            68: {1,1,7}
      8: {1,1,1}     41: {13}            71: {20}
     11: {5}         42: {1,2,4}         72: {1,1,1,2,2}
     12: {1,1,2}     43: {14}            73: {21}
     13: {6}         44: {1,1,5}         75: {2,3,3}
     17: {7}         45: {2,2,3}         76: {1,1,8}
     18: {1,2,2}     47: {15}            78: {1,2,6}
     19: {8}         48: {1,1,1,1,2}     79: {22}
     20: {1,1,3}     50: {1,3,3}         80: {1,1,1,1,3}
     23: {9}         52: {1,1,6}         83: {23}
     27: {2,2,2}     53: {16}            89: {24}
     28: {1,1,4}     59: {17}            92: {1,1,9}

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], OddQ[PrimeOmega[#]]&&IntegerQ[altprod[primeMS[#]]]&]

Crossrefs

The reciprocal version is A000290.

Allowing any alternating product <= 1 gives A001105.

Allowing any alternating product gives A026424.

Factorizations of this type are counted by A347441.

These partitions are counted by A347444.

Allowing any length gives A347454.

Allowing any alternating product > 1 gives A347465.

A027193 counts odd-length partitions.

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.

A344606 counts alternating permutations of prime indices.

A347446 counts partitions with integer alternating product.

A347457 ranks partitions with integer alt product, complement A347455.

A347461 counts possible alternating products of partitions.

A347462 counts possible reverse-alternating products of partitions.

Cf. A001222, A028260, A028982, A028983, A339890, A344617, A344653, A345958, A346703, A346704, A347437, A347443, A347450, A347451.

Sequence in context: A026422 A026424 A298207 * A229125 A228853 A141832

Adjacent sequences: A347450 A347451 A347452 * A347454 A347455 A347456

Keyword

nonn

Author

Gus Wiseman, Sep 24 2021

Status

approved