A188169 The number of divisors d of n of the form d == 1 (mod 8).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2

Offset

1,9

Comments

a(n) >= 1 as the divisor d=1 is always counted.

The largest terms up to n = 10^6 are each equal to 24. Those 8 terms are for n = 675675, 765765, 799425, 855855, 863379, 883575, 945945, or 987525. - Harvey P. Dale, May 31 2017

From David A. Corneth, Apr 06 2021: (Start)

a(n) can be computed from the prime factorization of n. Let v(n) = (n1, n3, n5, n7) where n_r is the number of divisors of n in class r (mod 8) (we do not care about even remainders). Then if gcd(k, m) = 1 we have v(k) = (k1, k3, k5, k7) so a(k) = k1, v(m) = (m1, m3, m5, m7) so a(m) = k1.

We have a(k*m) = (km)_1 = k1*m1 + k2*m2 + k3*m3 + k4*m4. The other (km)_3..(km)_7 have a similar expression.

If p == 1 (mod 8) then a(p^e) = e + 1 otherwise floor(e/2) + 1. (End)

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

David A. Corneth, PARI program.

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

a(n) + A188171(n) = A001826(n).

G.f.: Sum_{k>=1} x^k/(1 - x^(8*k)). - Ilya Gutkovskiy, Sep 11 2019

a(k) = a(2*k). - David A. Corneth, Apr 06 2021

Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,8) - (1 - gamma)/8 = A256781 - (1 - A001620)/8 = 0.735783... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

Maple

sigmamr := proc(n, m, r) local a, d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d, m) = r then a := a+1 ; end if; end do: a; end proc:
A188169 := proc(n) sigmamr(n, 8, 1) ; end proc:

Mathematica

Table[Count[Divisors[n], _?(Mod[#, 8]==1&)], {n, 100}] (* Harvey P. Dale, May 31 2017 *)

Prog

(PARI) a(n) = {my(d = divisors(n)); #select(x -> x%8 == 1, d)} \\ David A. Corneth, Apr 06 2021
(PARI) See PARI link \\ David A. Corneth, Apr 06 2021

Crossrefs

Cf. A001826, A188170, A188171, A188172.

Cf. A001620, A256781.

Sequence in context: A031279 A124778 A037831 * A107039 A249771 A030615

Adjacent sequences: A188166 A188167 A188168 * A188170 A188171 A188172

Keyword

nonn,easy

Author

R. J. Mathar, Mar 23 2011

Status

approved