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

0,9

Comments

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

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.

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

Example

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.

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

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 *)

Prog

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

Crossrefs

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

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

Sequence in context: A187447 A146292 A139039 * A206491 A359997 A122172

Adjacent sequences: A279058 A279059 A279060 * A279062 A279063 A279064

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Dec 05 2016

Status

approved