A188171 -id:A188171 - cp's OEIS frontend

A002325 Glaisher's J numbers.

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

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

Original entry on oeis.org

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

R. J. Mathar, Mar 23 2011

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)

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.

Cf. A001826, A188170, A188171, A188172.

Cf. A001620, A256781.

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:

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

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

\\ See PARI link. David A. Corneth, Apr 06 2021

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

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

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

A343109 Numbers having no divisor of the form 8*k + 5.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 34, 36, 38, 41, 43, 44, 46, 47, 48, 49, 51, 54, 56, 57, 59, 62, 64, 66, 67, 68, 71, 72, 73, 76, 79, 81, 82, 83, 86, 88, 89, 92, 94, 96, 97, 98, 99, 102, 103, 107, 108, 112, 113

Offset: 1

Jianing Song, Apr 05 2021

Numbers not divisible by at least one of 5, 13, 21, ...

			9 is a term since it has no divisor congruent to 5 modulo 8.

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), this sequence (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 0 in A188171.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343109(n) = (res(n,8,5) == 0)

A343113 Numbers having exactly 1 divisor of the form 8*k + 5.

Original entry on oeis.org

5, 10, 13, 15, 20, 21, 25, 26, 29, 30, 35, 37, 39, 40, 42, 50, 52, 53, 55, 58, 60, 61, 63, 69, 70, 74, 75, 77, 78, 80, 84, 87, 91, 93, 95, 100, 101, 104, 106, 109, 110, 111, 115, 116, 120, 122, 126, 133, 138, 140, 141, 143, 147, 148, 149, 150, 154, 155

Offset: 1

Jianing Song, Apr 05 2021

			52 is a term since among the divisors of 52 (namely 1, 2, 4, 13, 26 and 52), the only divisor congruent to 5 modulo 8 is 13.

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), A343111 (m=2, i=1), A343112 (m=1, i=3), this sequence (m=1, i=5), A141164 (m=1, i=7).
Indices of 1 in A188171.
A007521 is a subsequence.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343113(n) = (res(n,8,5) == 1)

A343106 Smallest number having exactly n divisors of the form 8*k + 5.

Original entry on oeis.org

1, 5, 45, 315, 585, 2205, 2925, 14175, 9945, 17325, 28665, 178605, 45045, 190575, 240975, 143325, 135135, 3189375, 225225, 93002175, 405405, 1403325, 1715175, 2401245, 675675, 3583125, 3239775, 1576575, 3468465, 94918019805, 2027025, 15436575, 2297295, 11609325, 16769025, 27286875, 3828825, 42879375, 117661005

Offset: 0

Jianing Song, Apr 05 2021

nonn

Smallest index of n in A188171.

a(n) exists for all n, since 5*3^(2n-2) has exactly n divisors of the form 8*k + 1, namely 5*3^0, 5*3^2, ..., 5*3^(2n-2). This actually gives an upper bound (which is too far from reality when n is large) for a(n).

			a(4) = 585 since it is the smallest number with exactly 4 divisors congruent to 5 modulo 8, namely 5, 13, 45 and 585.

Bert Dobbelaere, Table of n, a(n) for n = 0..200

Smallest number having exactly n divisors of the form 8*k + i: A343104 (i=1), A343105 (i=3), this sequence (i=5), A188226 (i=7).

Cf. A188171.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
a(n) = for(k=1, oo, if(res(k,8,5)==n, return(k)))

a(2n-1) <= 3^(2n-2) * 35, since 3^(2n-2) * 35 has exactly 2n-1 divisors congruent to 5 modulo 8: 3^0 * 5, 3^2 * 5, ..., 3^(2n-2) * 5, 3^1 * 7, 3^3 * 7, ..., 3^(2n-3) * 7.

a(2n) <= 3^(n-1) * 455, since 3^(n-1) * 455 has exactly 2n divisors congruent to 5 modulo 8: 3^0 * 5, 3^2 * 5, ..., 3^b * 5, 3^1 * 7, 3^3 * 7, ..., 3^a * 7, 3^0 * 13, 3^2 * 13, ..., 3^b * 13, 3^1 * 455, 3^3 * 455, ... 3^a * 455, where a is the largest odd number <= n-1 and b is the largest even number <= n-1.

More terms from Bert Dobbelaere, Apr 09 2021

cp's OEIS Frontend

A343136 Positions of records in A188171.

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A002325 Glaisher's J numbers.

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PARI

PARI

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

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PARI

PARI

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

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PARI

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

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PARI

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

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