A333695 Numerators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).
1, 3, 7, 11, 21, 7, 43, 43, 61, 63, 111, 77, 157, 129, 49, 171, 273, 61, 343, 231, 43, 333, 507, 301, 521, 471, 547, 473, 813, 147, 931, 683, 259, 819, 129, 671, 1333, 1029, 1099, 903, 1641, 43, 1807, 111, 427, 1521, 2163, 399, 2101, 1563, 637, 1727, 2757, 547, 2331
Offset: 1
Examples
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, ...
Crossrefs
Programs
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Mathematica
nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest Table[Sum[EulerPhi[n/d]/d, {d, Divisors[n]}], {n, 55}] // Numerator Table[Sum[1/GCD[n, k], {k, n}], {n, 55}] // Numerator Table[DivisorSigma[2, n^2]/(n DivisorSigma[1, n^2]), {n, 55}] // Numerator -
PARI
a(n) = numerator(sumdiv(n, d, eulerphi(n/d) / d)); \\ Michel Marcus, Apr 03 2020
Formula
a(n) = numerator of Sum_{d|n} phi(n/d) / d.
a(n) = numerator of Sum_{k=1..n} 1 / gcd(n,k).
a(n) = numerator of sigma_2(n^2) / (n * sigma_1(n^2)).
a(p) = p^2 - p + 1 where p is prime.
From Amiram Eldar, Nov 21 2022: (Start)
a(n) = numerator(A057660(n)/n).