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Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.
15

%I #7 Aug 09 2021 11:21:32

%S 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,

%T 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,

%U 59,6,61,2,21,8,5,22,67,34,3,14,71,12,73,2,15,38

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

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

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

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

%e 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.

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

%Y Positions of 2's are A001747.

%Y Positions of primes are A037143 (complement: A033942).

%Y The even reverse version appears to be A329888.

%Y Positions of first appearances are A342768.

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

%Y The reverse version is A346701.

%Y The even version is A346704.

%Y A001221 counts distinct prime factors.

%Y A001222 counts all prime factors.

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

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

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

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

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 08 2021