%I #12 Aug 13 2024 09:10:34

%S 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,

%T 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,

%U 31,3,8,13,3,1,2,23,5,1,6,1,37,5,2,11,3,1,4,9

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

%H Robert Israel, <a href="/A346704/b346704.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%e 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.

%p f:= proc(n) local F,i;

%p F:= ifactors(n)[2];

%p F:= sort(map(t -> t[1]$t[2],F));

%p mul(F[i],i=2..nops(F),2)

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Aug 12 2024

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

%Y Positions of first appearances are A129597.

%Y Positions of 1's are A008578.

%Y Positions of primes are A168645.

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

%Y The odd version is A346703 (sum: A346697).

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

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

%Y A001221 counts distinct prime factors.

%Y A001222 counts all prime factors.

%Y A027187 counts partitions of even length, ranked by A028260.

%Y A056239 adds up prime indices, row sums of A112798.

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

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

%Y A335433/A335448 rank separable/inseparable partitions.

%Y A344606 counts alternating permutations of prime indices.

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

%Y A346633 adds up the even bisection of standard compositions.

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

%K nonn,look

%O 1,4

%A _Gus Wiseman_, Aug 08 2021