A276736 a(n) = numerator of Sum_{d|n} tau(d)/d.

1, 2, 5, 11, 7, 10, 9, 13, 2, 14, 13, 55, 15, 18, 7, 57, 19, 4, 21, 77, 15, 26, 25, 65, 38, 30, 58, 99, 31, 14, 33, 15, 65, 38, 9, 11, 39, 42, 25, 91, 43, 30, 45, 13, 14, 50, 49, 95, 66, 76, 95, 165, 55, 116, 91, 117, 35, 62, 61, 77, 63, 66, 18, 247, 21, 130

Offset: 1

Author

Jaroslav Krizek, Sep 16 2016

Keywords

• nonn
• frac

Comments

Also numerators 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) = 10.
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) = 10.
		

Links

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

Crossrefs

Cf. A000005, A007429, A276737 (denominators).

Programs

Magma

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

Mathematica

Table[Numerator@ Total[DivisorSigma[0, #]/#] &@ Divisors@ n, {n, 66}] (* Michael De Vlieger, Sep 16 2016 *)

PARI

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

Formula

For all n we have: 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) * A276737(n) / a(n).