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
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
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Mar 23 2011
STATUS
approved