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A361565 a(n) is the numerator of the median of divisors of n.

%I A361565 #20 Mar 18 2023 15:26:06
%S A361565 1,3,2,2,3,5,4,3,3,7,6,7,7,9,4,4,9,9,10,9,5,13,12,5,5,15,6,11,15,11,
%T A361565 16,6,7,19,6,6,19,21,8,13,21,13,22,15,7,25,24,7,7,15,10,17,27,15,8,15,
%U A361565 11,31,30,8,31,33,8,8,9,17,34,21,13,17,36,17,37,39,10
%N A361565 a(n) is the numerator of the median of divisors of n.
%H A361565 Winston de Greef, <a href="/A361565/b361565.txt">Table of n, a(n) for n = 1..10000</a>
%H A361565 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%F A361565 a(n) = numerator((A033676(n) + A033677(n))/2).
%F A361565 a(p) = (1 + p)/2 if p is an odd prime.
%F A361565 For p a prime, a(p^k) = (p^((k-1)/2) + p^((k+1)/2))/2 if k is odd, a(p^k) = p^(k/2) if k is even.
%e A361565 a(9) = 3 since the divisors of 9 are 1, 3, 9, and their median is 3.
%e A361565 a(12) = 7 since the divisors of 12 are 1, 2, 3, 4, 6, 12, and their median is 7/2.
%t A361565 a[n_]:=Numerator[Median[Divisors[n]]]; Array[a,75]
%o A361565 (PARI) a(n) = my(d=divisors(n), m=#d+1); numerator((d[m\2] + d[m-m\2])/2); \\ _Michel Marcus_, Mar 16 2023
%Y A361565 Cf. A027750, A033676, A033677, A071090, A139710, A139711, A361566 (denominator).
%K A361565 nonn,frac
%O A361565 1,2
%A A361565 _Stefano Spezia_, Mar 15 2023