A361201 Product of the right half (exclusive) of the multiset of prime factors of n; a(1) = 0.

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

Offset: 1

Author

Gus Wiseman, Mar 10 2023

Keywords

• nonn
• look

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of 1's are A000040.

Positions of first appearances are A123666.

The left inclusive version A347043.

The inclusive version is A347044.

The left version is A361200.

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, A096825, A347045, A347046, A360005.

Programs

Maple

f:= proc(n) local F;
  F:= ifactors(n)[2];
  F:= sort(map(t -> t[1]$t[2],F));
  convert(F[ceil(nops(F)/2)+1 ..-1],`*`)
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Aug 12 2024

Mathematica

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

Formula

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

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