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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.

%I A361725 #21 May 19 2025 23:44:03
%S A361725 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,
%T A361725 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,
%U A361725 59,3,61,31,3,2,13,3,67,2,23,5,71,2,73,37,5,2,11,3
%N 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.
%H A361725 Robert Israel, <a href="/A361725/b361725.txt">Table of n, a(n) for n = 2..10000</a>
%F A361725 a(n) = A027746(n, floor(A001222(n)/2) + 1).
%F A361725 a(n) = 2*A361632(n)/A361633(n) - A079879(n) if A001222(n) is even.
%F A361725 a(n) = A361632(n) if A001222(n) is odd.
%e A361725 a(30) = a(2*3*5) = 3; a(60) = a(2*2*3*5) = 3; a(72) = a(2*2*2*3*3) = 2.
%p A361725 f:= proc(n) local F,m;
%p A361725   F:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));
%p A361725   m:= ceil((1+nops(F))/2);
%p A361725   F[m]
%p A361725 end proc:
%p A361725 map(f, [$2..100]); # _Robert Israel_, May 19 2025
%t A361725 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 *)
%Y A361725 Cf. A001222, A027746, A079879, A361632, A361633.
%K A361725 nonn
%O A361725 2,1
%A A361725 _Stefano Spezia_, Mar 22 2023