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Product of the left half (exclusive) of the multiset of prime factors of n; a(1) = 0.
17

%I #16 Nov 02 2024 09:13:36

%S 0,1,1,2,1,2,1,2,3,2,1,2,1,2,3,4,1,2,1,2,3,2,1,4,5,2,3,2,1,2,1,4,3,2,

%T 5,4,1,2,3,4,1,2,1,2,3,2,1,4,7,2,3,2,1,6,5,4,3,2,1,4,1,2,3,8,5,2,1,2,

%U 3,2,1,4,1,2,3,2,7,2,1,4,9,2,1,4,5,2,3

%N Product of the left half (exclusive) of the multiset of prime factors of n; a(1) = 0.

%H Amiram Eldar, <a href="/A361200/b361200.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) * A347044(n) = n.

%F A361201(n) * A347043(n) = n.

%F a(n) = Product_{k=1..floor(A001222(n)/2)} A027746(n,k) for n >= 2. - _Amiram Eldar_, Nov 02 2024

%e The prime factors of 250 are {2,5,5,5}, with left half (exclusive) {2,5}, with product 10, so a(250) = 10.

%t Table[If[n==1,0,Times@@Take[Join@@ConstantArray@@@FactorInteger[n],Floor[PrimeOmega[n]/2]]],{n,100}]

%t a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]}, Times @@ p[[1 ;; Floor[Length[p]/2]]]]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Nov 02 2024 *)

%Y Positions of 1's are A000040.

%Y Positions of 2's are A037143.

%Y The inclusive version is A347043.

%Y The right inclusive version A347044.

%Y The right version is A361201.

%Y A000005 counts divisors.

%Y A001221 counts distinct prime factors.

%Y A006530 gives greatest prime factor.

%Y A112798 lists prime indices, length A001222, sum A056239.

%Y A360616 gives half of bigomega (exclusive), inclusive A360617.

%Y A360673 counts multisets by right sum (exclusive), inclusive A360671.

%Y First for prime indices, second for partitions, third for prime factors:

%Y - A360676 gives left sum (exclusive), counted by A360672, product A361200.

%Y - A360677 gives right sum (exclusive), counted by A360675, product A361201.

%Y - A360678 gives left sum (inclusive), counted by A360675, product A347043.

%Y - A360679 gives right sum (inclusive), counted by A360672, product A347044.

%Y Cf. A001248, A026424, A027746, A096825, A347045, A347046, A360005.

%K nonn

%O 1,4

%A _Gus Wiseman_, Mar 10 2023