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

%I A346704 #12 Aug 13 2024 09:10:34
%S A346704 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 A346704 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 A346704 31,3,8,13,3,1,2,23,5,1,6,1,37,5,2,11,3,1,4,9
%N A346704 Product of primes at even positions in the weakly increasing list (with multiplicity) of prime factors of n.
%H A346704 Robert Israel, <a href="/A346704/b346704.txt">Table of n, a(n) for n = 1..10000</a>
%F A346704 a(n) * A346703(n) = n.
%F A346704 A056239(a(n)) = A346698(n).
%e A346704 The prime factors of 108 are (2,2,3,3,3), with even bisection (2,3), with product 6, so a(108) = 6.
%e A346704 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 A346704 f:= proc(n) local F,i;
%p A346704   F:= ifactors(n)[2];
%p A346704   F:= sort(map(t -> t[1]$t[2],F));
%p A346704   mul(F[i],i=2..nops(F),2)
%p A346704 end proc:
%p A346704 map(f, [$1..100]); # _Robert Israel_, Aug 12 2024
%t A346704 Table[Times@@Last/@Partition[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]],2],{n,100}]
%Y A346704 Positions of first appearances are A129597.
%Y A346704 Positions of 1's are A008578.
%Y A346704 Positions of primes are A168645.
%Y A346704 The sum of prime indices of a(n) is A346698(n).
%Y A346704 The odd version is A346703 (sum: A346697).
%Y A346704 The odd reverse version is A346701 (sum: A346699).
%Y A346704 The reverse version appears to be A329888 (sum: A346700).
%Y A346704 A001221 counts distinct prime factors.
%Y A346704 A001222 counts all prime factors.
%Y A346704 A027187 counts partitions of even length, ranked by A028260.
%Y A346704 A056239 adds up prime indices, row sums of A112798.
%Y A346704 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A346704 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A346704 A335433/A335448 rank separable/inseparable partitions.
%Y A346704 A344606 counts alternating permutations of prime indices.
%Y A346704 A344617 gives the sign of the alternating sum of prime indices.
%Y A346704 A346633 adds up the even bisection of standard compositions.
%Y A346704 Cf. A026424, A035363, A209281, A236913, A342768, A344653, A345957, A345958, A345960, A345961, A345962.
%K A346704 nonn,look
%O A346704 1,4
%A A346704 _Gus Wiseman_, Aug 08 2021