A361725 a(n) is the largest of two middle prime factors of n if the number of prime divisors counted with multiplicity (A001222(n)) is even, otherwise is the middle prime factor of n.

2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 2, 13, 7, 5, 2, 17, 3, 19, 2, 7, 11, 23, 2, 5, 13, 3, 2, 29, 3, 31, 2, 11, 17, 7, 3, 37, 19, 13, 2, 41, 3, 43, 2, 3, 23, 47, 2, 7, 5, 17, 2, 53, 3, 11, 2, 19, 29, 59, 3, 61, 31, 3, 2, 13, 3, 67, 2, 23, 5, 71, 2, 73, 37, 5, 2, 11, 3

Offset: 2

Author

Stefano Spezia, Mar 22 2023

Keywords

• nonn

Examples

			a(30) = a(2*3*5) = 3; a(60) = a(2*2*3*5) = 3; a(72) = a(2*2*2*3*3) = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A001222, A027746, A079879, A361632, A361633.

Programs

Maple

f:= proc(n) local F,m;
  F:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));
  m:= ceil((1+nops(F))/2);
  F[m]
end proc:
map(f, [$2..100]); # Robert Israel, May 19 2025

Mathematica

f[n_] := Block[{p = Flatten[Table[#1, {#2}] & @@@ FactorInteger@ n], len}, len = Length@ p; If[OddQ@ len, p[[(1 + len)/2]], p[[len/2+1]]]]; Table[f@ n, {n, 2, 78}] (* After Michael De Vlieger in A079879 *)

Formula

a(n) = A027746(n, floor(A001222(n)/2) + 1).

a(n) = 2*A361632(n)/A361633(n) - A079879(n) if A001222(n) is even.

a(n) = A361632(n) if A001222(n) is odd.