A347451 Numbers whose multiset of prime indices has integer reciprocal alternating product.

1, 2, 4, 6, 8, 9, 10, 14, 16, 18, 21, 22, 24, 25, 26, 32, 34, 36, 38, 39, 40, 46, 49, 50, 54, 56, 57, 58, 62, 64, 65, 72, 74, 81, 82, 84, 86, 87, 88, 90, 94, 96, 98, 100, 104, 106, 111, 115, 118, 121, 122, 126, 128, 129, 133, 134, 136, 142, 144, 146, 150, 152

Offset: 1

Author

Gus Wiseman, Sep 24 2021

Keywords

• nonn

Comments

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

Also Heinz numbers integer partitions with integer reverse-reciprocal 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: {}            32: {1,1,1,1,1}       65: {3,6}
      2: {1}           34: {1,7}             72: {1,1,1,2,2}
      4: {1,1}         36: {1,1,2,2}         74: {1,12}
      6: {1,2}         38: {1,8}             81: {2,2,2,2}
      8: {1,1,1}       39: {2,6}             82: {1,13}
      9: {2,2}         40: {1,1,1,3}         84: {1,1,2,4}
     10: {1,3}         46: {1,9}             86: {1,14}
     14: {1,4}         49: {4,4}             87: {2,10}
     16: {1,1,1,1}     50: {1,3,3}           88: {1,1,1,5}
     18: {1,2,2}       54: {1,2,2,2}         90: {1,2,2,3}
     21: {2,4}         56: {1,1,1,4}         94: {1,15}
     22: {1,5}         57: {2,8}             96: {1,1,1,1,1,2}
     24: {1,1,1,2}     58: {1,10}            98: {1,4,4}
     25: {3,3}         62: {1,11}           100: {1,1,3,3}
     26: {1,6}         64: {1,1,1,1,1,1}    104: {1,1,1,6}
		

Crossrefs

The version for reversed prime indices is A028982, counted by A119620.

The additive version is A119899, strict A028260.

Allowing any alternating product >= 1 gives A344609.

Factorizations of this type are counted by A347439.

Allowing any alternating product <= 1 gives A347450.

The non-reciprocal version is A347454.

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.

A347457 ranks partitions with integer alternating product.

Cf. A001222, A236913, A316523, A344617, A345958, A345959, A346703, A346704, A347437, A347446, A347455.

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[1/altprod[primeMS[#]]]&]