A279061 Number of divisors of n of the form 7*k + 1.
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, 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, 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
Offset: 0
Examples
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.
Links
- Robert Israel, Table of n, a(n) for n = 0..10000
- R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
Crossrefs
Programs
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Maple
N:= 200: # to get a(0)..a(N) V:= Vector(N): for k from 1 to N do R:= [seq(i,i=k..N,7*k)]; V[R]:= map(`+`,V[R],1); od: 0,seq(V[i],i=1..N); # Robert Israel, Dec 05 2016
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Mathematica
nmax = 120; CoefficientList[Series[Sum[x^k/(1 - x^(7 k)), {k, 1, nmax}], {x, 0, nmax}], x] nmax = 120; CoefficientList[Series[Sum[x^(7 k + 1)/(1 - x^(7 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x] Table[Count[Divisors[n],?(IntegerQ[(#-1)/7]&)],{n,0,100}] (* _Harvey P. Dale, Nov 08 2022 *) -
PARI
concat([0], Vec(sum(k=1, 100, x^k / (1 - x^(7*k))) + O(x^101))) \\ Indranil Ghosh, Mar 29 2017
Formula
G.f.: Sum_{k>=1} x^k/(1 - x^(7*k)).
G.f.: Sum_{k>=0} x^(7*k+1)/(1 - x^(7*k+1)).
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
Comments