This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A279060 #38 Jun 20 2025 01:29:06
%S A279060 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 A279060 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 A279060 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 A279060 Number of divisors of n of the form 6*k + 1.
%C A279060 Möbius transform is the period-6 sequence {1, 0, 0, 0, 0, 0, ...}.
%H A279060 Antti Karttunen, <a href="/A279060/b279060.txt">Table of n, a(n) for n = 0..65537</a>
%H A279060 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 A279060 G.f.: Sum_{k>=1} x^k/(1 - x^(6*k)).
%F A279060 G.f.: Sum_{k>=0} x^(6*k+1)/(1 - x^(6*k+1)).
%F A279060 From _Antti Karttunen_, Oct 03 2018: (Start)
%F A279060 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 A279060 a(n) = A035218(n) - A319995(n). (End)
%F A279060 a(n) = (A035218(n) + A035178(n)) / 2. - _David Radcliffe_, Jun 19 2025
%F A279060 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 A279060 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 A279060 nmax = 120; CoefficientList[Series[Sum[x^k/(1 - x^(6 k)), {k, 1, nmax}], {x, 0, nmax}], x]
%t A279060 nmax = 120; CoefficientList[Series[Sum[x^(6 k + 1)/(1 - x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
%t A279060 Table[Count[Divisors[n],_?(Mod[#,6]==1&)],{n,0,120}] (* _Harvey P. Dale_, Apr 27 2018 *)
%o A279060 (PARI) A279060(n) = if(!n,n,sumdiv(n, d, (1==(d%6)))); \\ _Antti Karttunen_, Jul 09 2017
%o A279060 (Python) from sympy import divisors
%o A279060 def A279060(n): return sum(d%6 == 1 for d in divisors(n)) # _David Radcliffe_, Jun 19 2025
%Y A279060 Cf. A001227, A001817, A001826, A001876, A035218, A140213, A188169, A319995, A320001.
%Y A279060 Cf. A001620, A016629, A222457 (psi(1/6)).
%K A279060 nonn,easy
%O A279060 0,8
%A A279060 _Ilya Gutkovskiy_, Dec 05 2016