A346704 Product of primes at even positions in the weakly increasing list (with multiplicity) of prime factors of n.

1, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 2, 1, 7, 5, 4, 1, 3, 1, 2, 7, 11, 1, 6, 5, 13, 3, 2, 1, 3, 1, 4, 11, 17, 7, 6, 1, 19, 13, 10, 1, 3, 1, 2, 3, 23, 1, 4, 7, 5, 17, 2, 1, 9, 11, 14, 19, 29, 1, 10, 1, 31, 3, 8, 13, 3, 1, 2, 23, 5, 1, 6, 1, 37, 5, 2, 11, 3, 1, 4, 9

Offset: 1

Author

Gus Wiseman, Aug 08 2021

Keywords

• nonn
• look

Examples

			The prime factors of 108 are (2,2,3,3,3), with even bisection (2,3), with product 6, so a(108) = 6.
The prime factors of 720 are (2,2,2,2,3,3,5), with even bisection (2,2,3), with product 12, so a(720) = 12.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of first appearances are A129597.

Positions of 1's are A008578.

Positions of primes are A168645.

The sum of prime indices of a(n) is A346698(n).

The odd version is A346703 (sum: A346697).

The odd reverse version is A346701 (sum: A346699).

The reverse version appears to be A329888 (sum: A346700).

A001221 counts distinct prime factors.

A001222 counts all prime factors.

A027187 counts partitions of even length, ranked by A028260.

A056239 adds up prime indices, row sums of A112798.

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

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.

Cf. A026424, A035363, A209281, A236913, A342768, A344653, A345957, A345958, A345960, A345961, A345962.

Programs

Maple

f:= proc(n) local F,i;
  F:= ifactors(n)[2];
  F:= sort(map(t -> t[1]$t[2],F));
  mul(F[i],i=2..nops(F),2)
end proc:
map(f, [$1..100]); # Robert Israel, Aug 12 2024

Mathematica

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

Formula

a(n) * A346703(n) = n.

A056239(a(n)) = A346698(n).