A324364 - OEIS

%I #15 Sep 08 2022 08:46:24

%S 1,2,3,8,5,6,7,64,27,25,11,432,13,98,25,1024,17,1944,19,4000,441,242,

%T 23,9216,125,169,729,14,29,11250,31,32768,363,289,1225,10077696,37,

%U 722,1521,256000,41,129654,43,21296,30375,1058,47,63700992,343,125000,867

%N a(n) = denominator of Sum_{d|n} sigma(d)/pod(d) where sigma(k) = the sum of the divisors of k (A000203) and pod(k) = the product of the divisors of k (A007955).

%C Sum_{d|n} sigma(d)/pod(d) > 1 for all n > 1.

%F a(p) = p for p = primes (A000040).

%F a(n) = n for n = 6 or when n is a noncomposite (in A008578).

%F a(n) = 1 for n = 1, ... (no other n <= 10^5).

%e For n=4; Sum_{d|4} sigma(d)/pod(d) = sigma(1)/pod(1) + sigma(2)/pod(2) + sigma(4)/pod(4) = 1/1 + 3/2 + 7/8 = 27/8; a(4) = 8.

%t Array[Denominator@ DivisorSum[#, Total[#]/(Times @@ #) &@ Divisors@ # &] &, 51] (* _Michael De Vlieger_, Feb 24 2019 *)

%o (Magma) [Denominator(&+[SumOfDivisors(d) / &*[c: c in Divisors(d)]: d in Divisors(n)]): n in [1..100]]

%o (PARI) a(n) = denominator(sumdiv(n, d, sigma(d)/vecprod(divisors(d)))); \\ _Michel Marcus_, Feb 23 2019

%Y Cf. A000040, A000203, A007955, A008578, A324363 (numerators).

%K nonn,frac

%O 1,2

%A _Jaroslav Krizek_, Feb 23 2019