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A324505 a(n) = numerator of Sum_{d|n} (d/pod(d)) where pod(k) = the product of the divisors of k (A007955).

%I A324505 #16 Jan 27 2025 14:26:57
%S A324505 1,2,2,5,2,19,2,21,7,31,2,529,2,43,46,169,2,1135,2,1441,64,67,2,52513,
%T A324505 11,79,64,2801,2,117001,2,2705,100,103,106,1122553,2,115,118,238561,2,
%U A324505 317521,2,6865,6886,139,2,20247937,15,8251,154,9569,2,557443,166
%N A324505 a(n) = numerator of Sum_{d|n} (d/pod(d)) where pod(k) = the product of the divisors of k (A007955).
%C A324505 Sum_{d|n} (d/pod(d)) >= 1 for all n >= 1.
%C A324505 Sum_{d|n} (d/pod(d)) = 2 iff n = primes (A000040).
%H A324505 Antti Karttunen, <a href="/A324505/b324505.txt">Table of n, a(n) for n = 1..20000</a>
%F A324505 a(p) = 2 for p = primes.
%e A324505 Sum_{d|n} (d/pod(d)) for n >= 1: 1, 2, 2, 5/2, 2, 19/6, 2, 21/8, 7/3, 31/10, 2, 529/144, 2, 43/14, 46/15, 169/64, ...
%e A324505 For n=4; Sum_{d|4} (d/pod(d)) = 1/pod(1) + 2/pod(2) + 4/pod(4) = 1/1 + 2/2 + 4/8 = 5/2; a(4) = 5.
%t A324505 Table[Numerator[Sum[k/Product[j, {j, Divisors[k]}], {k, Divisors[n]}]], {n, 1, 60}] (* _G. C. Greubel_, Mar 04 2019 *)
%o A324505 (Magma) [Numerator(&+[d / &*[c: c in Divisors(d)]: d in Divisors(n)]): n in [1..100]];
%o A324505 (Sage) [sum(k/product(j for j in k.divisors()) for k in n.divisors()).numerator() for n in (1..60)] # _G. C. Greubel_, Mar 04 2019
%o A324505 (PARI)
%o A324505 A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2));
%o A324505 A324505(n) = numerator(sumdiv(n,d,(d/A007955(d)))); \\ _Antti Karttunen_, Jan 27 2025
%Y A324505 Cf. A000040, A007955, A007956 (denominators).
%K A324505 nonn,frac
%O A324505 1,2
%A A324505 _Jaroslav Krizek_, Mar 03 2019