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A211177
Numerator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.
4
-1, 0, -1, 0, -1, 1, 1, 1, 1, 5, 19, 17, 29, 13, 21, 13, 47, 181, 503, 593, 533, 121, 1259, 1457, 6889, 7549, 7109, 7769, 52403, 59333, 11497, 6095, 29089, 61643, 59333, 63953, 62413, 7277, 21061, 2777, 10877, 11647, 3809, 3963, 1438, 271, 3064, 51439, 7217, 7493
OFFSET
1,10
LINKS
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq., Vol. 16 (2013) Article 13.6.3.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
FORMULA
a(n)/A211178(n) = c*log(n) + O(1) with a suitable constant c (see ref).
The constant above is c = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - Amiram Eldar, Nov 20 2020
More accurately, a(n)/A211178(n) ~ (A/3) * (log(n) + gamma - B - 8*log(2)/3) + O(log(n)^(5/3)/n), where A = zeta(2)*zeta(3)/zeta(6) (A082695), gamma is Euler's constant (A001620), and B = Sum_{p prime} log(p)/(p^2-p+1) (A085609) (Bordellès and Cloitre, 2013; Tóth, 2017). - Amiram Eldar, Oct 14 2022
EXAMPLE
Fractions begin with -1, 0, -1/2, 0, -1/4, 1/4, 1/12, 1/3, 1/6, 5/12, 19/60, 17/30, ...
MATHEMATICA
Numerator @ Accumulate[Table[(-1)^k/EulerPhi[k], {k, 1, 50}]] (* Amiram Eldar, Nov 20 2020 *)
PROG
(PARI) a(n)=numerator(sum(k=1, n, (-1)^k/eulerphi(k)))
CROSSREFS
Cf. A000010, A028415, A211178 (denominators).
Sequence in context: A109325 A370824 A169698 * A260645 A226658 A212153
KEYWORD
sign,frac
AUTHOR
Benoit Cloitre, Feb 01 2013
EXTENSIONS
More terms from Amiram Eldar, Nov 20 2020
STATUS
approved