cp's OEIS Frontend

A361200 Product of the left half (exclusive) of the multiset of prime factors of n; a(1) = 0.

%I A361200 #16 Nov 02 2024 09:13:36
%S A361200 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 A361200 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 A361200 3,2,1,4,1,2,3,2,7,2,1,4,9,2,1,4,5,2,3
%N A361200 Product of the left half (exclusive) of the multiset of prime factors of n; a(1) = 0.
%H A361200 Amiram Eldar, <a href="/A361200/b361200.txt">Table of n, a(n) for n = 1..10000</a>
%F A361200 a(n) * A347044(n) = n.
%F A361200 A361201(n) * A347043(n) = n.
%F A361200 a(n) = Product_{k=1..floor(A001222(n)/2)} A027746(n,k) for n >= 2. - _Amiram Eldar_, Nov 02 2024
%e A361200 The prime factors of 250 are {2,5,5,5}, with left half (exclusive) {2,5}, with product 10, so a(250) = 10.
%t A361200 Table[If[n==1,0,Times@@Take[Join@@ConstantArray@@@FactorInteger[n],Floor[PrimeOmega[n]/2]]],{n,100}]
%t A361200 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 A361200 Positions of 1's are A000040.
%Y A361200 Positions of 2's are A037143.
%Y A361200 The inclusive version is A347043.
%Y A361200 The right inclusive version A347044.
%Y A361200 The right version is A361201.
%Y A361200 A000005 counts divisors.
%Y A361200 A001221 counts distinct prime factors.
%Y A361200 A006530 gives greatest prime factor.
%Y A361200 A112798 lists prime indices, length A001222, sum A056239.
%Y A361200 A360616 gives half of bigomega (exclusive), inclusive A360617.
%Y A361200 A360673 counts multisets by right sum (exclusive), inclusive A360671.
%Y A361200 First for prime indices, second for partitions, third for prime factors:
%Y A361200 - A360676 gives left sum (exclusive), counted by A360672, product A361200.
%Y A361200 - A360677 gives right sum (exclusive), counted by A360675, product A361201.
%Y A361200 - A360678 gives left sum (inclusive), counted by A360675, product A347043.
%Y A361200 - A360679 gives right sum (inclusive), counted by A360672, product A347044.
%Y A361200 Cf. A001248, A026424, A027746, A096825, A347045, A347046, A360005.
%K A361200 nonn
%O A361200 1,4
%A A361200 _Gus Wiseman_, Mar 10 2023