A347454 Numbers whose multiset of prime indices has integer alternating product.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 108, 109, 112, 113

Offset: 1

Author

Gus Wiseman, Sep 26 2021

Keywords

• nonn

Comments

First differs from A265640 in having 42.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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

Also Heinz numbers of partitions with integer reverse-alternating product, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The terms and their prime indices begin:
      1: {}            20: {1,1,3}         47: {15}
      2: {1}           23: {9}             48: {1,1,1,1,2}
      3: {2}           25: {3,3}           49: {4,4}
      4: {1,1}         27: {2,2,2}         50: {1,3,3}
      5: {3}           28: {1,1,4}         52: {1,1,6}
      7: {4}           29: {10}            53: {16}
      8: {1,1,1}       31: {11}            59: {17}
      9: {2,2}         32: {1,1,1,1,1}     61: {18}
     11: {5}           36: {1,1,2,2}       63: {2,2,4}
     12: {1,1,2}       37: {12}            64: {1,1,1,1,1,1}
     13: {6}           41: {13}            67: {19}
     16: {1,1,1,1}     42: {1,2,4}         68: {1,1,7}
     17: {7}           43: {14}            71: {20}
     18: {1,2,2}       44: {1,1,5}         72: {1,1,1,2,2}
     19: {8}           45: {2,2,3}         73: {21}
		

Crossrefs

The even-length case is A000290.

The additive version is A026424.

Allowing any alternating product < 1 gives A119899, strict A028260.

Allowing any alternating product >= 1 gives A344609, multiplicative A347456.

Factorizations of this type are counted by A347437.

These partitions are counted by A347445, reverse A347446.

Allowing any alternating product <= 1 gives A347450.

The reciprocal version is A347451.

The odd-length case is A347453.

The version for reversed prime indices is A347457, complement A347455.

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

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.

A347461 counts possible alternating products of partitions.

A347462 counts possible reverse-alternating products of partitions.

Cf. A001105, A001222, A028982, A119620, A236913, A316523, A344653, A346703, A346704, A347443, A347439.

Programs

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[primeMS[#]]]&]