A276737 a(n) = denominator of Sum_{d|n} tau(d)/d.

1, 1, 3, 4, 5, 3, 7, 4, 1, 5, 11, 12, 13, 7, 3, 16, 17, 1, 19, 20, 7, 11, 23, 12, 25, 13, 27, 28, 29, 3, 31, 4, 33, 17, 5, 2, 37, 19, 13, 20, 41, 7, 43, 4, 5, 23, 47, 16, 49, 25, 51, 52, 53, 27, 55, 28, 19, 29, 59, 12, 61, 31, 7, 64, 13, 33, 67, 68, 69, 5, 71

Offset: 1

Author

Jaroslav Krizek, Sep 16 2016

Keywords

• nonn
• frac

Comments

Also denominator of (Sum_{d|n} sigma(d)) / n.

Examples

			For n=6; {d_6} = {1, 2, 3, 6}; {tau(d)_6} = {1, 2, 2, 4}; Sum_{d|6} tau(d)/d = 1/1 + 2/2 + 2/3 + 4/6 = 20/6 = 10/3; a(6) = 3.
For n=6; {d_6} = {1, 2, 3, 6}; {sigma(d)_6} = {1, 3, 4, 12};  (Sum_{d|6} sigma(d))/6 = (1+3+4+12)/6 = 10/3; a(6) = 3.
		

Links

• Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A007429, A068978, A276736 (numerators).

Programs

Magma

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

Mathematica

Table[Denominator@ Total[DivisorSigma[0, #]/#] &@ Divisors@ n, {n, 71}] (* Michael De Vlieger, Sep 16 2016 *)

PARI

a(n) = denominator(sumdiv(n, d, numdiv(d)/d)); \\ Michel Marcus, Sep 16 2016

Formula

a(A068978(n)) = 1.

For all n, n = (Sum_{d|n} sigma(d)) / (Sum_{d|n} tau(d)/d) = (Sum_{d|n} d*tau(n/d)) / (Sum_{d|n} tau(d)/d) = A007429(n) * a(n) / A276736(n).