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%I #34 Nov 25 2023 04:31:50
%S 0,1,1,1,1,1,1,2,1,1,1,1,1,2,2,1,1,1,1,2,1,2,1,1,1,2,2,1,2,1,1,2,1,1,
%T 1,2,1,2,2,2,1,1,2,2,1,1,1,1,1,3,2,1,2,1,1,2,2,2,1,1,1,2,2,2,1,2,1,2,
%U 1,1,2,1,1,2,2,2,2,2,2,2,1,1,1,1,2,2,2,1,1,1,1,4,1,2,1,2,1,2,3,1,2,1,1,2,2,2,1,1,1,2,2,2,2,1,2,2,1,2,1,2,1
%N Number of divisors of n of the form 6*k + 1.
%C Möbius transform is the period-6 sequence {1, 0, 0, 0, 0, 0, ...}.
%H Antti Karttunen, <a href="/A279060/b279060.txt">Table of n, a(n) for n = 0..65537</a>
%H R. A. Smith and M. V. Subbarao, <a href="https://doi.org/10.4153/CMB-1981-005-3">The average number of divisors in an arithmetic progression</a>, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
%F G.f.: Sum_{k>=1} x^k/(1 - x^(6*k)).
%F G.f.: Sum_{k>=0} x^(6*k+1)/(1 - x^(6*k+1)).
%F From _Antti Karttunen_, Oct 03 2018: (Start)
%F a(n) = A320001(n) + [1 == n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = 1 mod 6, and 0 otherwise.
%F a(n) = A035218(n) - A319995(n). (End)
%F Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,6) - (1 - gamma)/6 = 0.686263..., gamma(1,6) = -(psi(1/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023
%e a(14) = 2 because 14 has 4 divisors {1,2,7,14} among which 2 divisors {1,7} are of the form 6*k + 1.
%t nmax = 120; CoefficientList[Series[Sum[x^k/(1 - x^(6 k)), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 120; CoefficientList[Series[Sum[x^(6 k + 1)/(1 - x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
%t Table[Count[Divisors[n],_?(Mod[#,6]==1&)],{n,0,120}] (* _Harvey P. Dale_, Apr 27 2018 *)
%o (PARI) A279060(n) = if(!n,n,sumdiv(n, d, (1==(d%6)))); \\ _Antti Karttunen_, Jul 09 2017
%Y Cf. A001227, A001817, A001826, A001876, A035218, A140213, A188169, A319995, A320001.
%Y Cf. A001620, A016629, A222457 (psi(1/6)).
%K nonn,easy
%O 0,8
%A _Ilya Gutkovskiy_, Dec 05 2016
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