A276736 - OEIS

%I #21 Sep 08 2022 08:46:17

%S 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,

%T 30,58,99,31,14,33,15,65,38,9,11,39,42,25,91,43,30,45,13,14,50,49,95,

%U 66,76,95,165,55,116,91,117,35,62,61,77,63,66,18,247,21,130

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

%C Also numerators of (Sum_{d|n} sigma(d)) / n.

%H Jaroslav Krizek, <a href="/A276736/b276736.txt">Table of n, a(n) for n = 1..1000</a>

%F 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).

%e 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.

%e 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.

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

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

%o (PARI) a(n) = numerator(sumdiv(n, d, numdiv(d)/d)); \\ _Michel Marcus_, Sep 16 2016

%Y Cf. A000005, A007429, A276737 (denominators).

%K nonn,frac

%O 1,2

%A _Jaroslav Krizek_, Sep 16 2016