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A188169 The number of divisors d of n of the form d == 1 (mod 8). 12
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 (list; graph; refs; listen; history; text; internal format)
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

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

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.

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

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Last modified July 30 17:21 EDT 2021. Contains 346359 sequences. (Running on oeis4.)