A333695 - OEIS

%I #15 Nov 21 2022 09:39:19

%S 1,3,7,11,21,7,43,43,61,63,111,77,157,129,49,171,273,61,343,231,43,

%T 333,507,301,521,471,547,473,813,147,931,683,259,819,129,671,1333,

%U 1029,1099,903,1641,43,1807,111,427,1521,2163,399,2101,1563,637,1727,2757,547,2331

%N Numerators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

%F a(n) = numerator of Sum_{d|n} phi(n/d) / d.

%F a(n) = numerator of Sum_{k=1..n} 1 / gcd(n,k).

%F a(n) = numerator of sigma_2(n^2) / (n * sigma_1(n^2)).

%F a(p) = p^2 - p + 1 where p is prime.

%F From _Amiram Eldar_, Nov 21 2022: (Start)

%F a(n) = numerator(A057660(n)/n).

%F Sum_{k=1..n} a(k)/A333696(k) ~ c * n^2, where c = zeta(3)/(2*zeta(2)) = 0.365381... (A346602). (End)

%e 1, 3/2, 7/3, 11/4, 21/5, 7/2, 43/7, 43/8, 61/9, 63/10, 111/11, 77/12, 157/13, 129/14, 49/5, ...

%t nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest

%t Table[Sum[EulerPhi[n/d]/d, {d, Divisors[n]}], {n, 55}] // Numerator

%t Table[Sum[1/GCD[n, k], {k, n}], {n, 55}] // Numerator

%t Table[DivisorSigma[2, n^2]/(n DivisorSigma[1, n^2]), {n, 55}] // Numerator

%o (PARI) a(n) = numerator(sumdiv(n, d, eulerphi(n/d) / d)); \\ _Michel Marcus_, Apr 03 2020

%Y Cf. A000010, A000203, A001157, A018804, A057660, A071708, A072155, A074947, A074949, A333696 (denominators), A346602.

%K nonn,frac

%O 1,2

%A _Ilya Gutkovskiy_, Apr 02 2020