A309710 -id:A309710 - cp's OEIS frontend

A328895 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^2.

8, 7, 2, 3, 5, 8, 0, 2, 4, 9, 5, 4, 8, 5, 9, 9, 4, 1, 7, 6, 9, 6, 9, 5, 1, 1, 7, 0, 2, 1, 1, 7, 5, 6, 6, 1, 2, 3, 9, 9, 8, 3, 2, 8, 3, 8, 6, 8, 5, 0, 5, 2, 9, 5, 7, 6, 9, 1, 8, 7, 0, 8, 3, 4, 3, 9, 9, 8, 8, 4, 7, 0, 3, 5, 4, 1, 3, 4, 6, 5, 1, 8, 3, 3, 4, 2, 5, 1, 6, 7, 1

Offset: 0

Jianing Song, Nov 19 2019

Let Chi() be a primitive character modulo d, the so-called Dirichlet L-series L(s,Chi) is the analytic continuation (see the functional equations involving L(s,Chi) in the MathWorld link entitled Dirichlet L-Series) of the sum Sum_{k>=1} Chi(k)/k^s, Re(s)>0 (if d = 1, the sum converges requires Re(s)>1).
If s != 1, we can represent L(s,Chi) in terms of the Hurwitz zeta function by L(s,Chi) = (Sum_{k=1..d} Chi(k)*zeta(s,k/d))/d^s.
L(s,Chi) can also be represented in terms of the polylog function by L(s,Chi) = (Sum_{k=1..d} Chi'(k)*polylog(s,u^k))/(Sum_{k=1..d} Chi'(k)*u^k), where Chi' is the complex conjugate of Chi, u is any primitive d-th root of unity.
If m is a positive integer, we have L(m,Chi) = (Sum_{k=1..d} Chi(k)*polygamma(m-1,k/d))/((-d)^m*(m-1)!).
In this sequence we have Chi = A091337 and s = 2.

			1 - 1/3^2 - 1/5^2 + 1/7^2 + 1/9^2 - 1/11^2 - 1/13^2 + 1/15^2 + ... = Pi^2/(8*sqrt(2)) = 0.8723580249...

M. W. Coffey, Summatory relations and prime products for the Stieltjes constants and other related results, arXiv:1701.07064 [math.NT], 2017, eq. (2.1).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, Section 2.2 at m=8, r=2, s=2.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A091337, A346728, A346727, A346260.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^2, where d is a fundamental discriminant: A309710 (d=-8), A103133 (d=-7), A006752 (d=-4), A086724 (d=-3), A013661 (d=1), A328717 (d=5), this sequence (d=8), A258414 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^s: A196525 (s=1), this sequence (s=2), A329715 (s=3).

RealDigits[Pi^2/(8*Sqrt[2]), 10, 102] // First

default(realprecision, 100); Pi^2/(8*sqrt(2))

Equals Pi^2/(8*sqrt(2)).
Equals (zeta(2,1/8) - zeta(2,3/8) - zeta(2,5/8) + zeta(2,7/8))/64, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(2,u) - polylog(2,u^3) - polylog(2,-u) + polylog(2,-u^3))/sqrt(8), where u = sqrt(2)/2 + i*sqrt(2)/2 is an 8th primitive root of unity, i = sqrt(-1).
Equals (polygamma(1,1/8) - polygamma(1,3/8) - polygamma(1,5/8) + polygamma(1,7/8))/64.
Equals -Integral_{x=0..oo} log(x)/(x^4 + 1) dx. - Amiram Eldar, Jul 17 2020
Equals 1/(Product_{p prime == 1 or 7 (mod 8)} (1 - 1/p^2) * Product_{p prime == 3 or 5 (mod 8)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

A328717 Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^2.

Original entry on oeis.org

7, 0, 6, 2, 1, 1, 4, 0, 3, 2, 5, 9, 7, 4, 0, 9, 6, 9, 9, 3, 1, 0, 0, 3, 1, 7, 5, 7, 6, 2, 5, 6, 4, 0, 2, 7, 6, 6, 0, 2, 4, 6, 4, 7, 1, 8, 5, 2, 9, 4, 6, 8, 6, 3, 9, 4, 2, 1, 1, 7, 4, 0, 2, 1, 6, 5, 6, 7, 7, 6, 0, 4, 4, 3, 8, 3, 8, 3, 0, 0, 7, 6, 8, 3, 3, 7, 4, 5, 6, 6, 4

Offset: 0

Jianing Song, Nov 19 2019

Let Chi() be a primitive character modulo d, the so-called Dirichlet L-series L(s,Chi) is the analytic continuation (see the functional equations involving L(s,Chi) in the MathWorld link entitled Dirichlet L-Series) of the sum Sum_{k>=1} Chi(k)/k^s, Re(s)>0 (if d = 1, the sum converges requires Re(s)>1).
If s != 1, we can represent L(s,Chi) in terms of the Hurwitz zeta function by L(s,Chi) = (Sum_{k=1..d} Chi(k)*zeta(s,k/d))/d^s.
L(s,Chi) can also be represented in terms of the polylog function by L(s,Chi) = (Sum_{k=1..d} Chi'(k)*polylog(s,u^k))/(Sum_{k=1..d} Chi'(k)*u^k), where Chi' is the complex conjugate of Chi, u is any primitive d-th root of unity.
If m is a positive integer, we have L(m,Chi) = (Sum_{k=1..d} Chi(k)*polygamma(m-1,k/d))/((-d)^m*(m-1)!).
In this sequence we have Chi = A080891 and s = 2.

			1 - 1/2^2 - 1/3^2 + 1/4^2 + 1/6^2 - 1/7^2 - 1/8^2 + 1/9^2 + ... = 4*Pi^2/(25*sqrt(5)) = 0.70621140325974096993100317576256402766024647185294...

M. W. Coffey, Summatory relations and prime products for the Stieltjes constants and other related results, arXiv:1701.07064 [math.NT], 2017, eq. (2.1).
H.-J. Seiffert, Problem B-705, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 29, No. 4 (1991), p. 372; An Application of a Series Expansion for (arcsinx)^2, Solution to Problem B-705, ibid., Vol. 31, No. 1 (1993), pp. 85-86.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A000045, A001906, A002736, A080891.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^2, where d is a fundamental discriminant: A309710 (d=-8), A103133 (d=-7), A006752 (d=-4), A086724 (d=-3), A013661 (d=1), this sequence (d=5), A328895 (d=8), A258414 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^s: A086466 (s=1), this sequence (s=2), A328723 (s=3).

RealDigits[4*Pi^2/(25*Sqrt[5]), 10, 102] // First

default(realprecision, 100); 4*Pi^2/(25*sqrt(5))

Equals 4*Pi^2/(25*sqrt(5)).
Equals (zeta(2,1/5) - zeta(2,2/5) - zeta(2,3/5) + zeta(2,4/5))/25, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(2,u) - polylog(2,u^2) - polylog(2,u^3) + polylog(2,u^4))/sqrt(5), where u = exp(2*Pi*i/5) is a 5th primitive root of unity, i = sqrt(-1).
Equals (polygamma(1,1/5) - polygamma(1,2/5) - polygamma(1,3/5) - polygamma(1,4/5))/25.
Equals Sum_{k>=1} Fibonacci(2*k)/(k^2*binomial(2*k,k)) = Sum_{k>=1} A001906(k)/A002736(k) (Seiffert, 1991). - Amiram Eldar, Jan 17 2022
Equals 1/(Product_{p prime == 1 or 4 (mod 5)} (1 - 1/p^2) * Product_{p prime == 2 or 3 (mod 5)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

A088965 Number of solutions to x^2 + 2y^2 == 1 (mod n).

Original entry on oeis.org

1, 2, 2, 4, 6, 4, 8, 16, 6, 12, 10, 8, 14, 16, 12, 32, 16, 12, 18, 24, 16, 20, 24, 32, 30, 28, 18, 32, 30, 24, 32, 64, 20, 32, 48, 24, 38, 36, 28, 96, 40, 32, 42, 40, 36, 48, 48, 64, 56, 60, 32, 56, 54, 36, 60, 128, 36, 60, 58, 48, 62, 64, 48, 128, 84, 40, 66, 64, 48, 96

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 28 2003

Andrew Howroyd, Table of n, a(n) for n = 1..10000
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.

Cf. A087561, A060968, A309710.

[n eq 1 select 1 else #[x: x in [1..n], y in [1..n] | (x^2+2*y^2) mod n eq 1]: n in [1..80]]; // Vincenzo Librandi, Jul 16 2018

A088965 := proc(n) local a,x,y ; a := 0 ; for x from 0 to n-1 do for y from 0 to n-1 do if (x^2+2*y^2) mod n = 1 mod n then a := a+1 ; end if; end do; end do ; a ; end proc:
seq(A088965(n),n=1..70) ; # R. J. Mathar, Jan 07 2011

a[1]=1; a[n_]:=Length@Rest@Union@Flatten@Table[If[Mod[i^2 + 2 j^2, n]==1, i+I j, 0], {i, 0, n-1}, {j, 0, n-1}]; Table[a[n], {n, 1, 80}] (* Vincenzo Librandi, Jul 16 2018 *)

a(n)={my(v=vector(n)); for(i=0, n-1, v[i^2%n + 1]++); sum(i=0, n-1, v[i+1]*v[(1-2*i)%n + 1])} \\ Andrew Howroyd, Jul 09 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==2, 2^e*if(e>2, 2, 1), p^(e-1)*if(abs(p%8-2)==1, p-1, p+1)))} \\ Andrew Howroyd, Jul 09 2018

Multiplicative with a(2^e) = 2^e for e <= 2, a(2^e) = 2^(e + 1) for e > 2, a(p^e) = (p-1)*p^(e-1) for p-2 mod 8 = +-1, a(p^e) = (p+1)*p^(e-1) for p-2 mod 8 = +-3. - Andrew Howroyd, Jul 13 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = (9/(16*A309710)) = 0.528300880442971272... . - Amiram Eldar, Nov 21 2023

A087561 Number of solutions to x^2 + 2y^2 == 0 (mod n).

Original entry on oeis.org

1, 2, 5, 4, 1, 10, 1, 8, 21, 2, 21, 20, 1, 2, 5, 16, 33, 42, 37, 4, 5, 42, 1, 40, 25, 2, 81, 4, 1, 10, 1, 32, 105, 66, 1, 84, 1, 74, 5, 8, 81, 10, 85, 84, 21, 2, 1, 80, 49, 50, 165, 4, 1, 162, 21, 8, 185, 2, 117, 20, 1, 2, 21, 64, 1, 210, 133, 132, 5, 2, 1, 168, 145, 2, 125, 148, 21

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 24 2003

Andrew Howroyd, Table of n, a(n) for n = 1..10000
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.

Cf. A086933, A088964, A088965, A089002, A309710.

a[n_] := If[n == 1, 1, Product[{p, e} = pe; Which[p == 2, 2^e, Abs[Mod[p, 8] - 2] != 1, (p^2)^Quotient[e, 2], True, (p + e (p-1)) p^(e-1)], {pe, FactorInteger[n]}]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)

a(n)={my(v=vector(n)); for(i=0, n-1, v[i^2%n + 1]++); sum(i=0, n-1, v[i+1]*v[(-2*i)%n + 1])} \\ Andrew Howroyd, Jul 16 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 2^e, if(abs(p%8-2)<>1, (p^2)^(e\2), (p+e*(p-1))*p^(e-1))))} \\ Andrew Howroyd, Jul 16 2018

Multiplicative with a(2^e) = 2^e, a(p^e) = p^(2*floor(e/2)) for p - 2 == +-3 (mod 8), a(p^e) = ((p-1)*e+p)*p^(e-1) for p - 2 == +-1 (mod 8). - Andrew Howroyd, Jul 16 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi/(4*sqrt(2)*A309710) = 0.521595326207... . - Amiram Eldar, Nov 21 2023

More terms from David Wasserman, Jun 07 2005

A089002 Number of non-congruent solutions to x^2 + 2y^2 == -1 (mod n).

Original entry on oeis.org

1, 2, 2, 4, 6, 4, 8, 0, 6, 12, 10, 8, 14, 16, 12, 0, 16, 12, 18, 24, 16, 20, 24, 0, 30, 28, 18, 32, 30, 24, 32, 0, 20, 32, 48, 24, 38, 36, 28, 0, 40, 32, 42, 40, 36, 48, 48, 0, 56, 60, 32, 56, 54, 36, 60, 0, 36, 60, 58, 48, 62, 64, 48, 0, 84, 40, 66, 64, 48, 96

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 02 2003

Andrew Howroyd, Table of n, a(n) for n = 1..10000
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.

Cf. A088965, A086932, A309710.

f[2, e_] := If[e < 3, 2^e, 0]; f[p_, e_] := If[MemberQ[{1, 7}, Mod[p - 2, 8]], (p - 1), (p + 1)] * p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

a(n)={my(v=vector(n)); for(i=0, n-1, v[i^2%n + 1]++); sum(i=0, n-1, v[i+1]*v[(-1-2*i)%n + 1])} \\ Andrew Howroyd, Jul 09 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==2, if(e>2, 0, 2^e), p^(e-1)*if(abs(p%8-2)==1, p-1, p+1)))} \\ Andrew Howroyd, Jul 09 2018

Multiplicative with a(2^e) = 2^e for e <= 2, a(2^e) = 0 for e > 2, a(p^e) = (p-1)*p^(e-1) for p-2 mod 8 = +-1, a(p^e) = (p+1)*p^(e-1) for p-2 mod 8 = +-3. - Andrew Howroyd, Jul 15 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = 7/(16*A309710) = 0.410900684788977656... . - Amiram Eldar, Nov 21 2023

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A328895 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^2.

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A328717 Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^2.

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A088965 Number of solutions to x^2 + 2y^2 == 1 (mod n).

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A087561 Number of solutions to x^2 + 2y^2 == 0 (mod n).

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A089002 Number of non-congruent solutions to x^2 + 2y^2 == -1 (mod n).

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