A346703 Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.

1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 6, 13, 2, 3, 4, 17, 6, 19, 10, 3, 2, 23, 4, 5, 2, 9, 14, 29, 10, 31, 8, 3, 2, 5, 6, 37, 2, 3, 4, 41, 14, 43, 22, 15, 2, 47, 12, 7, 10, 3, 26, 53, 6, 5, 4, 3, 2, 59, 6, 61, 2, 21, 8, 5, 22, 67, 34, 3, 14, 71, 12, 73, 2, 15, 38

Offset: 1

Author

Gus Wiseman, Aug 08 2021

Keywords

• nonn

Examples

			The prime factors of 108 are (2,2,3,3,3), with odd bisection (2,3,3), with product 18, so a(108) = 18.
The prime factors of 720 are (2,2,2,2,3,3,5), with odd bisection (2,2,3,5), with product 60, so a(720) = 60.
		

Crossrefs

Positions of 2's are A001747.

Positions of primes are A037143 (complement: A033942).

The even reverse version appears to be A329888.

Positions of first appearances are A342768.

The sum of prime indices of a(n) is A346697(n), reverse: A346699.

The reverse version is A346701.

The even version is A346704.

A001221 counts distinct prime factors.

A001222 counts all prime factors.

A056239 adds up prime indices, row sums of A112798.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A209281 (shifted) adds up the odd bisection of standard compositions.

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

A335433/A335448 rank separable/inseparable partitions.

A344606 counts alternating permutations of prime indices.

A344617 gives the sign of the alternating sum of prime indices.

A346633 adds up the even bisection of standard compositions.

A346698 gives the sum of the even bisection of prime indices.

A346700 gives the sum of the even bisection of reversed prime indices.

Cf. A025047, A027187, A027193, A053738, A097805, A106356, A341446, A344653, A345957, A345958, A345959.

Programs

Mathematica

Table[Times@@First/@Partition[Append[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]],0],2],{n,100}]

Formula

a(n) * A346704(n) = n.

A056239(a(n)) = A346697(n).