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Product of the right half (exclusive) of the multiset of prime factors of n; a(1) = 0.
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%I #11 Aug 13 2024 09:10:30

%S 0,1,1,2,1,3,1,2,3,5,1,3,1,7,5,4,1,3,1,5,7,11,1,6,5,13,3,7,1,5,1,4,11,

%T 17,7,9,1,19,13,10,1,7,1,11,5,23,1,6,7,5,17,13,1,9,11,14,19,29,1,15,1,

%U 31,7,8,13,11,1,17,23,7,1,9,1,37,5,19,11,13,1

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

%H Robert Israel, <a href="/A361201/b361201.txt">Table of n, a(n) for n = 1..10000</a>

%F A361200(n) * A347044(n) = n.

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

%e The prime factors of 250 are {2,5,5,5}, with right half (exclusive) {5,5}, with product 25, so a(250) = 25.

%p f:= proc(n) local F;

%p F:= ifactors(n)[2];

%p F:= sort(map(t -> t[1]$t[2],F));

%p convert(F[ceil(nops(F)/2)+1 ..-1],`*`)

%p end proc:

%p f(1):= 0:

%p map(f, [$1..100]); # _Robert Israel_, Aug 12 2024

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

%Y Positions of 1's are A000040.

%Y Positions of first appearances are A123666.

%Y The left inclusive version A347043.

%Y The inclusive version is A347044.

%Y The left version is A361200.

%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, A096825, A347045, A347046, A360005.

%K nonn,look

%O 1,4

%A _Gus Wiseman_, Mar 10 2023