A347450 Numbers whose multiset of prime indices has alternating product <= 1.

1, 2, 4, 6, 8, 9, 10, 14, 15, 16, 18, 21, 22, 24, 25, 26, 32, 33, 34, 35, 36, 38, 39, 40, 46, 49, 50, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 69, 72, 74, 77, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 98, 100, 104, 106, 111, 115, 118, 119, 121, 122

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

Also Heinz numbers integer partitions with reverse-alternating product <= 1, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Also numbers whose multiset of prime indices has alternating sum <= 1.

Examples

			The initial terms and their prime indices:
      1: {}            26: {1,6}           56: {1,1,1,4}
      2: {1}           32: {1,1,1,1,1}     57: {2,8}
      4: {1,1}         33: {2,5}           58: {1,10}
      6: {1,2}         34: {1,7}           60: {1,1,2,3}
      8: {1,1,1}       35: {3,4}           62: {1,11}
      9: {2,2}         36: {1,1,2,2}       64: {1,1,1,1,1,1}
     10: {1,3}         38: {1,8}           65: {3,6}
     14: {1,4}         39: {2,6}           69: {2,9}
     15: {2,3}         40: {1,1,1,3}       72: {1,1,1,2,2}
     16: {1,1,1,1}     46: {1,9}           74: {1,12}
     18: {1,2,2}       49: {4,4}           77: {4,5}
     21: {2,4}         50: {1,3,3}         81: {2,2,2,2}
     22: {1,5}         51: {2,7}           82: {1,13}
     24: {1,1,1,2}     54: {1,2,2,2}       84: {1,1,2,4}
     25: {3,3}         55: {3,5}           85: {3,7}
		

Crossrefs

The additive version (alternating sum <= 0) is A028260.

The reverse version is A028982, counted by A119620.

Allowing any alternating product < 1 gives A119899.

Factorizations of this type are counted by A339846, complement A339890.

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

Partitions of this type are counted by A347443.

Allowing any integer alternating product gives A347454, reciprocal A347451.

The complement is A347465, reverse A028983, counted by A347448.

A056239 adds up prime indices, row sums of A112798.

A236913 counts partitions of 2n with reverse-alternating sum <= 0.

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 lists Heinz numbers of partitions with integer alternating product.

Cf. A001105, A001222, A027193, A344617, A345958, A346703, A346704, A347449, A347461, A347462.

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],altprod[primeMS[#]]<=1&]

Formula

Union of A028982 and A119899.

Union of A028260 and A001105.