A188169 -id:A188169 - cp's OEIS frontend

A343134 Positions of records in A188169.

1, 9, 81, 153, 891, 1377, 3825, 11025, 15147, 31977, 95931, 187425, 287793, 675675, 1091475, 1576575, 1686825, 2027025, 2297295, 3828825, 6185025, 11486475, 18555075, 26801775, 34459425, 43648605, 72747675, 117515475, 218243025, 352546425, 509233725, 654729075, 1003917915

Offset: 1

David A. Corneth, Apr 06 2021

nonn

			153 is in the sequence as A188169(153) = 4 via the divisors 1, 9, 17 and 153 and no positive integer < 153 has at least four such (1 (mod 8)) divisors.

Cf. A188169, A343104.

A002325 Glaisher's J numbers.

Original entry on oeis.org

1, 1, 2, 1, 0, 2, 0, 1, 3, 0, 2, 2, 0, 0, 0, 1, 2, 3, 2, 0, 0, 2, 0, 2, 1, 0, 4, 0, 0, 0, 0, 1, 4, 2, 0, 3, 0, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0, 2, 1, 1, 4, 0, 0, 4, 0, 0, 4, 0, 2, 0, 0, 0, 0, 1, 0, 4, 2, 2, 0, 0, 0, 3, 2, 0, 2, 2, 0, 0, 0, 0, 5, 2, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 1, 6, 1, 0, 4, 0, 0, 0

Offset: 1

N. J. A. Sloane

Number of integer solutions to the equation x^2 + 2*y^2 = n when (-x, -y) and (x, y) are counted as the same solution.
For n nonzero, a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -8. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			x + x^2 + 2*x^3 + x^4 + 2*x^6 + x^8 + 3*x^9 + 2*x^11 + 2*x^12 + x^16 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(iii).
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 19.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).
J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91. [Incomplete annotated scanned copy]
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 5.
N. J. A. Sloane, Transforms.
Index entries for sequences related to Glaisher's numbers.

Cf. A033715, A093954.
Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

S:= series( (JacobiTheta3(0,q)*JacobiTheta3(0,q^2)-1)/2, q, 1001):
seq(coeff(S,q,j), j=1..1000); # Robert Israel, Dec 01 2015

a[n_] := Total[ KroneckerSymbol[-8, #] & /@ Divisors[n]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Nov 25 2011, after Michael Somos *)
QP = QPochhammer; s = ((QP[q^2]^3*QP[q^4]^3)/(QP[q]^2*QP[q^8]^2)-1)/(2q) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

a(n) = if( n<1, 0, issquare(n)-issquare(2*n) + 2*sum(i=1,sqrtint(n\2), issquare(n-2*i^2)))

{a(n) = if( n<1, 0, qfrep([ 1, 0; 0, 2],n)[n])} \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker( -2, p) * X))[n])} \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -2, d)))} \\ Michael Somos, Aug 23 2005

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p%8<4, e+1, !(e%2))))))} \\ Michael Somos, Oct 23 2006

{a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^3 * eta(x^4 + A)^3 * eta(x^8 + A)^-2, n) / 2)}

a(n) = my(f=factor(n>>valuation(n,2)), e); prod(i=1, #f~, e=f[i, 2]; if( f[i, 1]%8<4, e+1, 1 - e%2)) \\ Charles R Greathouse IV, Sep 09 2014

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s) + Kronecker(m, p)*p^(-2s))^(-1) for m = -2.
Moebius transform is period 8 sequence [ 1, 0, 1, 0, -1, 0, -1, 0, ...]. - Michael Somos, Aug 23 2005
G.f.: (theta_3(q) * theta_3(q^2) - 1) / 2 = Sum_{k>0} Kronecker( -2, n) * x^k / (1 - x^k) = Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).
Multiplicative with a(2^e) = 1, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8). - Michael Somos, Oct 23 2006
A033715(n) = 2 * a(n) unless n=0.
a(n) = A188169(n) + A188170(n) - A188171(n) - A188172(n) [Hirschhorn]. - R. J. Mathar, Mar 23 2011
G.f.: A(x) = 2*(1+x^2)/(G(0)-2*x*(1+x^2)); G(k) = 1+x+x^(2*k)*(1+x^3+x^(2*k+1)+x^(2*k+4)+x^(4*k+3)+x^(4*k+4)) - x*(1+x^(2*k))*(1+x^(2*k+4))*(1+x^(4*k+4))^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 03 2012
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(2)) = 1.110720... (A093954). - Amiram Eldar, Oct 11 2022

A035185 Number of divisors of n == 1 or 7 (mod 8) minus number of divisors of n == 3 or 5 (mod 8).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 0, 2, 0, 1, 2, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 3, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 0, 0, 2, 1, 2, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 3, 0, 1, 0, 0, 2, 0, 0

Offset: 1

N. J. A. Sloane

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = 2.
Let zetaQ(sqrt(2))(s) = Sum (1/(Z(sqrt(2)):A)^s), a Dedekind zeta function, where A runs through the nonzero ideals of Z(sqrt(2)) and where (Z(sqrt(2)):A) is the norm of A; then zetaQ(sqrt(2))(s) = Sum_{n>=1}, a(n)/n^s); Sum{k=1..n} a(k) is asymptotic to c*n where c = log(1 + sqrt(2))/sqrt(2). - Benoit Cloitre, Jan 01 2003
Inverse Moebius transform of A091337.
a(n) is the number of solutions to the equation n = x^2 - 2*y^2 in integers where -x < 2*y <= x. [Uspensky and Heaslet] - Michael Somos, Feb 17 2020
Coefficients of Dedekind zeta function for the quadratic number field of discriminant 8. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			G.f. = x + x^2 + x^4 + 2*x^7 + x^8 + x^9 + 2*x^14 + x^16 + 2*x^17 + x^18 + ...
a(7) = 2 because 7 = 3^2 - 2*(+1)^2 = 3^2 - 2*(-1)^2. - _Michael Somos_, Feb 17 2020

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 368.

G. C. Greubel, Table of n, a(n) for n = 1..10000
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.

Cf. A035251, A078462, A188169, A188170, A188171, A188172, A259829.
Cf. A002193, A091648.
Moebius transform gives A091337.
Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ 2, #] &]]; (* Michael Somos, Jul 06 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ x^k (1 - x^(2 k)) / (1 + x^(4 k)), {k, n}], {x, 0, n}]; (* Michael Somos, Jul 06 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # <= 2, 1, Mod[#, 8] > 1 && Mod[#, 8] < 7, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* Michael Somos, Jul 06 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker(2, d)))};

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker(2, p)*X)))[n])};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%8>1 && p%8<7, !(e%2), e+1)))}; \\ Michael Somos, Aug 17 2006

{a(n) = if( n<1, 0, polcoeff( sum(k=1, n, x^k * (1 - x^(2*k)) / (1 + x^(4*k)), x * O(x^n)), n))}; \\ Michael Somos, Jul 06 2015

G.f.: Sum_{k>0} x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
-(-1)^(n*(n-1)/2)*a(n) = Sum_{n >= 1} (-1)^n * q^(n*(n+1)/2)*(1-q)*(1-q^2)*...*(1-q^(n-1))/ ((1+q)*(1+q^2)*...*(1+q^n)). - Jeremy Lovejoy, Jun 12 2009
a(n) = (-1)^floor(n/2) * A259829(n). - Michael Somos, Jul 06 2015
a(n) is multiplicative with a(2^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if p == 3, 5 (mod 8), a(p^e) = e + 1 if p == 1, 7 (mod 8). - Jianing Song, Sep 07 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(sqrt(2)+1)/sqrt(2) = A091648/A002193 = 0.623225... . - Amiram Eldar, Oct 11 2022

A279061 Number of divisors of n of the form 7*k + 1.

Original entry on oeis.org

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

Ilya Gutkovskiy, Dec 05 2016

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

			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.

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.

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

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

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

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 *)

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

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

A279060 Number of divisors of n of the form 6*k + 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1

Offset: 0

Ilya Gutkovskiy, Dec 05 2016

Möbius transform is the period-6 sequence {1, 0, 0, 0, 0, 0, ...}.

			a(14) = 2 because 14 has 4 divisors {1,2,7,14} among which 2 divisors {1,7} are of the form 6*k + 1.

Antti Karttunen, Table of n, a(n) for n = 0..65537
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.

Cf. A001227, A001817, A001826, A001876, A035218, A140213, A188169, A319995, A320001.

Cf. A001620, A016629, A222457 (psi(1/6)).

nmax = 120; CoefficientList[Series[Sum[x^k/(1 - x^(6 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 120; CoefficientList[Series[Sum[x^(6 k + 1)/(1 - x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Count[Divisors[n],?(Mod[#,6]==1&)],{n,0,120}] (* _Harvey P. Dale, Apr 27 2018 *)

A279060(n) = if(!n,n,sumdiv(n, d, (1==(d%6)))); \\ Antti Karttunen, Jul 09 2017

from sympy import divisors
def A279060(n): return sum(d%6 == 1 for d in divisors(n)) # David Radcliffe, Jun 19 2025

G.f.: Sum_{k>=1} x^k/(1 - x^(6*k)).
G.f.: Sum_{k>=0} x^(6*k+1)/(1 - x^(6*k+1)).
From Antti Karttunen, Oct 03 2018: (Start)
a(n) = A320001(n) + [1 == n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = 1 mod 6, and 0 otherwise.
a(n) = A035218(n) - A319995(n). (End)
a(n) = (A035218(n) + A035178(n)) / 2. - David Radcliffe, Jun 19 2025
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,6) - (1 - gamma)/6 = 0.686263..., gamma(1,6) = -(psi(1/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A188170 The number of divisors d of n of the form d == 3 (mod 8).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 1

Offset: 1

R. J. Mathar, Mar 23 2011

a(3n) >= 1 as the divisor d=3 contributes to the count then.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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.

Cf. A001842, A002325, A188169, A188171, A188172.

Cf. A001620, A256782.

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:
A188170 := proc(n) sigmamr(n,8,3) ; end proc:

Table[Count[Divisors[n],?(Mod[#,8]==3&)],{n,100}] (* _Harvey P. Dale, Jul 08 2013 *)

a(n) = sumdiv(n, d, (d%8) == 3); \\ Michel Marcus, Nov 05 2018

a(n) + A188172(n) = A001842(n).
A188169(n) + a(n) - A188171(n) - A188172(n) = A002325(n).
G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(8*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,8) - (1 - gamma)/8 = A256782 - (1 - A001620)/8 = 0.0314716... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A188171 The number of divisors d of n of the form d == 5 (mod 8).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0, 2

Offset: 1

R. J. Mathar, Mar 23 2011

a(5n) >= 1 as d=5 contributes to the count.

			a(13) = 1 because the divisor d=13 is 8+5 == 5 (mod 8).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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.

Cf. A001826, A002325, A188169, A188170, A188172.

Cf. A001620, A016631, A354634 (psi(5/8)).

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:
A188171 := proc(n) sigmamr(n,8,5) ; end proc:

a[n_] := DivisorSum[n, 1 &, Mod[#, 8] == 5 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A188171(n) = sumdiv(n, d, (5==(d%8)));  \\ Antti Karttunen, Jul 09 2017

A188169(n)+a(n) = A001826(n).
A188169(n)+A188170(n)-a(n)-A188172(n) = A002325(n).
G.f.: Sum_{k>=1} x^(5*k)/(1 - x^(8*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,8) - (1 - gamma)/8 = -0.131189..., gamma(5,8) = -(psi(5/8) + log(8))/8 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A188172 Number of divisors d of n of the form d == 7 (mod 8).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 1

R. J. Mathar, Mar 23 2011

			a(A007522(i)) = 1, any i.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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.

Cf. A001842, A002325, A004771, A141164, A188169, A188170, A188171, A188226.

Cf. A001620, A016631, A354635 (psi(7/8)).

a188172 n = length $ filter ((== 0) . mod n) [7,15..n]
-- Reinhard Zumkeller, Mar 26 2011

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:
A188172 := proc(n) sigmamr(n,8,7) ; end proc:

Table[Count[Divisors[n],?(Mod[#,8]==7&)],{n,90}] (* _Harvey P. Dale, Mar 08 2014 *)

a(n) = sumdiv(n, d, (d % 8) == 7); \\ Amiram Eldar, Nov 25 2023

A188170(n)+a(n) = A001842(n).
A188169(n)+A188170(n)-A188171(n)-a(n) = A002325(n).
a(A188226(n))=n and a(m)<>n for m<A188226(n), n>=0; a(A141164(n))=1. - Reinhard Zumkeller, Mar 26 2011
G.f.: Sum_{k>=1} x^(7*k)/(1 - x^(8*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(7,8) - (1 - gamma)/8 = -0.212276..., gamma(7,8) = -(psi(7/8) + log(8))/8 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A343107 Numbers having exactly 1 divisor of the form 8k + 1, that is, numbers with no divisor of the form 8k + 1 other than 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 35, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 52, 53, 55, 56, 58, 59, 60, 61, 62, 64, 67, 69, 70, 71, 74, 76, 77, 78, 79, 80, 83, 84, 86, 87, 88, 91, 92, 93, 94, 95, 96

Offset: 1

Jianing Song, Apr 05 2021

Numbers not divisible by at least one of 9, 17, 25, ...

			7 is a term since it has no divisor congruent to 1 modulo 8 other than 1.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: this sequence (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).

Indices of 1 in A188169.

Select[Range[100],NoneTrue[Rest[Divisors[#]],Mod[#,8]==1&]&] (* Harvey P. Dale, Jun 01 2022 *)

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343107(n) = (res(n,8,1) == 1)

A343111 Numbers having exactly 2 divisors of the form 8k + 1, that is, numbers with exactly 1 divisor of the form 8k + 1 other than 1.

Original entry on oeis.org

9, 17, 18, 25, 27, 33, 34, 36, 41, 45, 49, 50, 51, 54, 57, 63, 65, 66, 68, 72, 73, 75, 82, 85, 89, 90, 97, 98, 100, 102, 105, 108, 113, 114, 117, 119, 121, 123, 125, 126, 129, 130, 132, 135, 136, 137, 144, 145, 146, 147, 150, 161, 164, 165, 169, 170, 175

Offset: 1

Jianing Song, Apr 05 2021

			63 is a term since among the divisors of 63 (namely 1, 3, 7, 9, 21 and 63), the only divisors congruent to 1 modulo 8 are 1 and 9.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), this sequence (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).
Indices of 2 in A188169.
A007519 is a subsequence.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343111(n) = (res(n,8,1) == 2)

cp's OEIS Frontend

A343134 Positions of records in A188169.

Views

Author

Keywords

Examples

Crossrefs

A002325 Glaisher's J numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

PARI

PARI

Formula

A035185 Number of divisors of n == 1 or 7 (mod 8) minus number of divisors of n == 3 or 5 (mod 8).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Formula

A279061 Number of divisors of n of the form 7*k + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A279060 Number of divisors of n of the form 6*k + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A188170 The number of divisors d of n of the form d == 3 (mod 8).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A343107 Numbers having exactly 1 divisor of the form 8k + 1, that is, numbers with no divisor of the form 8k + 1 other than 1.

A343111 Numbers having exactly 2 divisors of the form 8k + 1, that is, numbers with exactly 1 divisor of the form 8k + 1 other than 1.