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

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

Offset: 1

Author

Gus Wiseman, Mar 10 2023

Keywords

• nonn

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of 1's are A000040.

Positions of 2's are A037143.

The inclusive version is A347043.

The right inclusive version A347044.

The right version is A361201.

A000005 counts divisors.

A001221 counts distinct prime factors.

A006530 gives greatest prime factor.

A112798 lists prime indices, length A001222, sum A056239.

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

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

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

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

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

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

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

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

Programs

Mathematica

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

Formula

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

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

a(n) = Product_{k=1..floor(A001222(n)/2)} A027746(n,k) for n >= 2. - Amiram Eldar, Nov 02 2024