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A324982 a(n) = numerator of Sum_{d|n} (pod(d)/tau(d)) where pod(k) = the product of the divisors of k (A007955) and tau(k) = the number of the divisors of k (A000005).

%I A324982 #19 Feb 28 2025 06:01:03
%S A324982 1,2,5,14,7,25,9,62,23,59,13,1819,15,109,245,3382,19,1987,21,2731,465,
%T A324982 257,25,250747,271,355,775,22295,31,405385,33,28434,1121,599,1253,
%U A324982 6726169,39,745,1557,642763,43,1556549,45,28657,61031,1085,49,765671783,713
%N A324982 a(n) = numerator of Sum_{d|n} (pod(d)/tau(d)) where pod(k) = the product of the divisors of k (A007955) and tau(k) = the number of the divisors of k (A000005).
%C A324982 Sum_{d|n} (pod(d)/tau(d)) > 1 for all n > 1.
%H A324982 Robert Israel, <a href="/A324982/b324982.txt">Table of n, a(n) for n = 1..10000</a>
%F A324982 a(p) = p + 2 for p = odd primes.
%e A324982 Sum_{d|n} (pod(d)/tau(d)) for n >= 1: 1, 2, 5/2, 14/3, 7/2, 25/2, 9/2, 62/3, 23/2, 59/2, ...
%e A324982 For n=4; Sum_{d|4} (pod(d)/tau(d)) = pod(1)/tau(1) + pod(2)/tau(2) + pod(4)/tau(4) = 1/1 + 2/2 + 8/3 = 14/3;  a(4) = 14.
%t A324982 Array[Numerator@ DivisorSum[#, Apply[Times, Divisors@ #]/DivisorSigma[0, #] &] &, 49] (* _Michael De Vlieger_, Mar 24 2019 *)
%o A324982 (Magma) [Numerator(&+[&*[c: c in Divisors(d)] / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]];
%o A324982 (PARI) a(n) = numerator(sumdiv(n, d, vecprod(divisors(d))/numdiv(d))); \\ _Michel Marcus_, Mar 23 2019
%Y A324982 Cf. A000203, A007955, A324983 (denominators).
%K A324982 nonn,frac,look
%O A324982 1,2
%A A324982 _Jaroslav Krizek_, Mar 22 2019