A324500 a(n) = denominator of Sum_{d|n} sigma(d)/tau(d) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of divisors of k (A000005).

1, 2, 1, 6, 1, 2, 1, 12, 3, 1, 1, 2, 1, 2, 1, 60, 1, 3, 1, 3, 1, 2, 1, 4, 3, 1, 3, 6, 1, 1, 1, 60, 1, 1, 1, 9, 1, 2, 1, 3, 1, 2, 1, 6, 3, 2, 1, 20, 1, 6, 1, 3, 1, 3, 1, 12, 1, 1, 1, 1, 1, 2, 3, 420, 1, 2, 1, 3, 1, 1, 1, 18, 1, 1, 1, 6, 1, 1, 1, 15, 15, 1, 1, 2

Offset: 1

Author

Jaroslav Krizek, Mar 02 2019

Keywords

• nonn
• frac

Comments

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

Sum_{d|n} sigma(d)/tau(d) = n only for numbers n = 1, 3, 10 and 30.

Examples

			Sum_{d|n} sigma(d)/tau(d) for n >= 1: 1, 5/2, 3, 29/6, 4, 15/2, 5, 103/12, 22/3, 10, 7, 29/2, 8, 25/2, 12, 887/60, ...
For n=4; Sum_{d|4} sigma(d)/tau(d) = sigma(1)/tau(1) + sigma(2)/tau(2) + sigma(4)/tau(4) = 1/1 + 3/2 + 7/3 = 29/6;  a(4) = 6.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000005, A000203, A323779, A323780, A323781, A324499 (numerators), A306639.

Programs

Magma

[Denominator(&+[SumOfDivisors(d) / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]

Mathematica

Table[Denominator[Sum[DivisorSigma[1, k]/DivisorSigma[0, k], {k, Divisors[n]}]], {n, 1, 100}] (* G. C. Greubel, Mar 04 2019 *)

PARI

a(n) = denominator(sumdiv(n, d, sigma(d)/numdiv(d))); \\ Michel Marcus, Mar 03 2019

Sage

[sum(sigma(k,1)/sigma(k,0) for k in n.divisors() ).denominator() for n in (1..100)] # G. C. Greubel, Mar 04 2019

Formula

a(p) = 1 for odd primes p.

a(n) = 1 for numbers in A306639.