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A279061
Number of divisors of n of the form 7*k + 1.
13
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
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
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Dec 05 2016
STATUS
approved