A324364 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).

1, 2, 3, 8, 5, 6, 7, 64, 27, 25, 11, 432, 13, 98, 25, 1024, 17, 1944, 19, 4000, 441, 242, 23, 9216, 125, 169, 729, 14, 29, 11250, 31, 32768, 363, 289, 1225, 10077696, 37, 722, 1521, 256000, 41, 129654, 43, 21296, 30375, 1058, 47, 63700992, 343, 125000, 867

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..51.

Formula

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

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

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

Example

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.

Mathematica

Array[Denominator@ DivisorSum[#, Total[#]/(Times @@ #) &@ Divisors@ # &] &, 51] (* Michael De Vlieger, Feb 24 2019 *)

Prog

(Magma) [Denominator(&+[SumOfDivisors(d) / &*[c: c in Divisors(d)]: d in Divisors(n)]): n in [1..100]]
(PARI) a(n) = denominator(sumdiv(n, d, sigma(d)/vecprod(divisors(d)))); \\ Michel Marcus, Feb 23 2019

Crossrefs

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

Sequence in context: A110340 A183091 A308360 * A265344 A089489 A284047

Adjacent sequences: A324361 A324362 A324363 * A324365 A324366 A324367

Keyword

nonn,frac

Author

Jaroslav Krizek, Feb 23 2019

Status

approved