A279061 - OEIS

%I #28 Nov 25 2023 04:27:30

%S 0,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,1,1,1,1,2,1,2,1,1,1,1,2,2,1,2,1,

%T 1,1,2,1,1,1,2,1,1,2,2,2,1,1,2,1,2,1,1,1,1,1,2,2,2,1,2,1,1,1,3,1,2,1,

%U 1,1,1,2,3,1,1,2,1,1,2,1,2,1,1,1,1,2,2,2,3,1,2,1,2,1,1,1,2,1,1,2

%N Number of divisors of n of the form 7*k + 1.

%C Möebius transform is a period-7 sequence {1, 0, 0, 0, 0, 0, 0, ...}.

%H Robert Israel, <a href="/A279061/b279061.txt">Table of n, a(n) for n = 0..10000</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^(7*k)).

%F G.f.: Sum_{k>=0} x^(7*k+1)/(1 - x^(7*k+1)).

%F Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,7) - (1 - gamma)/7 = 0.713612..., gamma(1,7) = -(psi(1/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023

%e a(8) = 2 because 8 has 4 divisors {1,2,4,8} among which 2 divisors {1,8} are of the form 7*k + 1.

%p N:= 200: # to get a(0)..a(N)

%p V:= Vector(N):

%p for k from 1 to N do

%p   R:= [seq(i,i=k..N,7*k)];

%p   V[R]:= map(`+`,V[R],1);

%p od:

%p 0,seq(V[i],i=1..N); # _Robert Israel_, Dec 05 2016

%t nmax = 120; CoefficientList[Series[Sum[x^k/(1 - x^(7 k)), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 120; CoefficientList[Series[Sum[x^(7 k + 1)/(1 - x^(7 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

%t Table[Count[Divisors[n],_?(IntegerQ[(#-1)/7]&)],{n,0,100}] (* _Harvey P. Dale_, Nov 08 2022 *)

%o (PARI) concat([0], Vec(sum(k=1, 100, x^k / (1 - x^(7*k))) + O(x^101))) \\ _Indranil Ghosh_, Mar 29 2017

%Y Cf. A001227, A001817, A001826, A001876, A188169.

%Y Cf. A001620, A016630, A354627 (psi(1/7)).

%K nonn,easy

%O 0,9

%A _Ilya Gutkovskiy_, Dec 05 2016