cp's OEIS Frontend

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

%I A346703 #7 Aug 09 2021 11:21:32
%S A346703 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 A346703 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 A346703 59,6,61,2,21,8,5,22,67,34,3,14,71,12,73,2,15,38
%N A346703 Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.
%F A346703 a(n) * A346704(n) = n.
%F A346703 A056239(a(n)) = A346697(n).
%e A346703 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 A346703 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 A346703 Table[Times@@First/@Partition[Append[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]],0],2],{n,100}]
%Y A346703 Positions of 2's are A001747.
%Y A346703 Positions of primes are A037143 (complement: A033942).
%Y A346703 The even reverse version appears to be A329888.
%Y A346703 Positions of first appearances are A342768.
%Y A346703 The sum of prime indices of a(n) is A346697(n), reverse: A346699.
%Y A346703 The reverse version is A346701.
%Y A346703 The even version is A346704.
%Y A346703 A001221 counts distinct prime factors.
%Y A346703 A001222 counts all prime factors.
%Y A346703 A056239 adds up prime indices, row sums of A112798.
%Y A346703 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A346703 A209281 (shifted) adds up the odd bisection of standard compositions.
%Y A346703 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A346703 A335433/A335448 rank separable/inseparable partitions.
%Y A346703 A344606 counts alternating permutations of prime indices.
%Y A346703 A344617 gives the sign of the alternating sum of prime indices.
%Y A346703 A346633 adds up the even bisection of standard compositions.
%Y A346703 A346698 gives the sum of the even bisection of prime indices.
%Y A346703 A346700 gives the sum of the even bisection of reversed prime indices.
%Y A346703 Cf. A025047, A027187, A027193, A053738, A097805, A106356, A341446, A344653, A345957, A345958, A345959.
%K A346703 nonn
%O A346703 1,2
%A A346703 _Gus Wiseman_, Aug 08 2021