A175629 -id:A175629 - cp's OEIS frontend

A103133 Decimal expansion of Dirichlet series L_{-7}(2).

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

Offset: 1

Eric W. Weisstein, Jan 23 2005

			1.151925470544491047...

D. Cvijovic, Proof of the Borwein-Broadhurst conjecture for a dilogarithmic integral arising in quantum field theory, arXiv:1011.0195 [math-ph], 2010.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 98-99.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, L(m=7,r=4,s=2).
Eric Weisstein's World of Mathematics, Dirichlet L-Series.

Cf. A326919, A327135.

(PolyGamma[1, 1/7] + PolyGamma[1, 2/7] - PolyGamma[1, 3/7] + PolyGamma[1, 4/7] - PolyGamma[1, 5/7] - PolyGamma[1, 6/7])/49 // RealDigits[#, 10, 102]& // First

(Psi(1, 1/7) + Psi(1, 2/7) - Psi(1, 3/7) + Psi(1, 4/7) - Psi(1, 5/7) - Psi(1, 6/7))/49, where Psi(1, x) is the polygamma function of order 1.
Equals Sum_{n>=1} A175629(n)/n^2. - R. J. Mathar, Jan 15 2021
Equals 1/(Product_{p prime == 1, 2 or 4 (mod 7)} (1 - 1/p^2) * Product_{p prime == 3, 5 or 6 (mod 7)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

Formula updated by Jean-François Alcover, Apr 01 2015

A322829 a(n) = Jacobi (or Kronecker) symbol (n/21).

Original entry on oeis.org

0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0

Offset: 0

Jianing Song, Dec 27 2018

Period 21: repeat [0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1].
Also a(n) = Kronecker symbol (21/n).
This sequence is one of the three non-principal real Dirichlet characters modulo 21. The other two are Jacobi or Kronecker symbols {(n/63)} (or {(-63/n)}) and {(n/147)} (or {(-147/n)}).

Eric Weisstein's World of Mathematics, Kronecker Symbol
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1,0,-1,0,1,-1,0,1,-1).

Cf. A035203 (inverse Moebius transform).

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017(d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), this sequence (d=21), A322796 (d=24).

JacobiSymbol[Range[0, 100], 21] (* Paolo Xausa, Mar 19 2025 *)

PARI
```
a(n) = kronecker(n, 21)
```

a(n) = 1 for n == 1, 4, 5, 16, 17, 20 (mod 21); -1 for n == 2, 8, 10, 11, 13, 19 (mod 21); 0 for n that are not coprime with 21.
Completely multiplicative with a(p) = a(p mod 21) for primes p.
a(n) = A102283(n)*A175629(n).
a(n) = a(n+21) = -a(n) for all n in Z.
From Chai Wah Wu, Feb 18 2021: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-6) + a(n-8) - a(n-9) + a(n-11) - a(n-12) for n > 11.
G.f.: -x*(x - 1)*(x + 1)*(x^8 - 2*x^7 + 2*x^6 + 2*x^2 - 2*x + 1)/(x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1). (End)

A126837 Ramanujan numbers (A000594) read mod 7^2.

Original entry on oeis.org

1, 25, 7, 47, 28, 28, 14, 4, 37, 14, 22, 35, 21, 7, 0, 31, 28, 43, 0, 42, 0, 11, 46, 28, 32, 35, 28, 21, 23, 0, 0, 31, 7, 14, 0, 24, 46, 0, 0, 14, 14, 0, 37, 5, 7, 23, 42, 21, 0, 16, 0, 7, 36, 14, 28, 7, 0, 36, 42, 0, 14, 0, 28, 7, 0, 28, 15, 42, 28, 0, 2, 1, 14, 23, 28, 0, 14, 0, 39, 35, 46

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Oddmund Kolberg, Note on Ramanujan's Function tau(n), Mathematica Scandinavica, Vol. 10 (1962), pp. 171-172; alternative link.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

Cf. A000594, A013957, A126836 (mod 7^1), this sequence (mod 7^2), A126838 (mod 7^3).

a[n_] := Mod[RamanujanTau[n], 49]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 49; \\ Amiram Eldar, Jan 05 2025

a(n) == n * sigma_9(n) (mod 7^2) if Legendre symbol (n,7) = A175629(n) = -1 (Kolberg, 1962). - Amiram Eldar, Jan 05 2025

A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

Original entry on oeis.org

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

Offset: 1

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 = A175629 and s = 1.

			1 + 1/2 - 1/3 + 1/4 - 1/5 - 1/6 + 1/8 + 1/9 - 1/10 + 1/11 - 1/12 - 1/13 + ... = Pi/sqrt(7) = 1.1874104117...

Sergio A. Carrillo, Where did the examples of Abel's continuity theorem go?, arXiv:2010.10290 [math.HO], 2020. See p. 11.
Eric Weisstein's World of Mathematics, Class Number.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A175629.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k, where d is a fundamental discriminant: A093954 (d=-8), this sequence (d=-7), A003881 (d=-4), A073010 (d=-3), A086466 (d=5), A196525 (d=8), A196530 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^s: this sequence (s=1), A103133 (s=2), A327135 (s=3).

RealDigits[Pi/Sqrt[7], 10, 102] // First

PARI
```
default(realprecision, 100); Pi/sqrt(7)
```

Equals Pi/sqrt(7). This is related to the class number formula: if d<0 is the fundamental discriminant of an imaginary quadratic number field, Chi(k) = Kronecker(d,k), then L(1,Chi) = Sum_{k>=1} Kronecker(d,k)/k = 2*Pi*h(d)/(sqrt(|d|)*w(d)), where h(d) is the class number of K = Q[sqrt(d)], w(d) is the number of elements in K whose norms are 1 (w(d) = 6 if d = -3, 4 if d = -4 and 2 if d < -4). Here d = -7, h(d) = 1, w(d) = 2.
Equals (polylog(1,u) + polylog(1,u^2) - polylog(1,u^3) + polylog(1,u^4) - polylog(1,u^5) - polylog(1,u^6))/sqrt(-7), where u = exp(2*Pi*i/7) is a 7th primitive root of unity, i = sqrt(-1).
Equals (polygamma(0,1/7) + polygamma(0,2/7) - polygamma(0,3/7) + polygamma(0,4/7) - polygamma(0,5/7) - polygamma(0,6/7))/49.
Equals 1/Product_{p prime} (1 - Kronecker(-7,p)/p), where Kronecker(-7,p) = 0 if p = 7, 1 if p == 1, 2 or 4 (mod 7) or -1 if p == 3, 5 or 6 (mod 7). - Amiram Eldar, Dec 17 2023

A091395 a(n) = Product_{ p | n } (1 + Legendre(-7,p) ).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 02 2004

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A091379, A175629.

with(numtheory); L := proc(n,N) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul((1+legendre(N,t1[i][1])),i=1..nops(t1)); end; [seq(L(n,-7),n=1..120)];

a[n_] := Times@@ (1+KroneckerSymbol[-7, #]& /@ FactorInteger[n][[All, 1]]);
Array[a, 105] (* Jean-François Alcover, Apr 08 2020 *)

a(n)={my(f=factor(n)[,1]); prod(i=1, #f, 1 + kronecker(-7, f[i]))} \\ Andrew Howroyd, Jul 23 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 7*sqrt(7)/(8*Pi) = 0.736897... . - Amiram Eldar, Oct 17 2022

A323378 Square array read by antidiagonals: T(n,k) = Kronecker symbol (-n/k), n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 0, -1, 1, -1, 1, 1, 1, 0, 0, 0, 1, 1, -1, -1, 1, -1, -1, 1, 0, 1, 0, -1, 0, -1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 0, -1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, 1, -1

Offset: 1

Jianing Song, Jan 12 2019

If A215200 is arranged into a square array A215200(n,k) = kronecker symbol(n/k) with n >= 0, k >= 1, then this sequence gives the other half of the array.

Note that there is no such n such that the n-th row and the n-th column are the same.

			Table begins
  1,  1, -1,  1,  1, -1, -1,  1,  1,  1, ... ((-1/k) = A034947)
  1,  0,  1,  0, -1,  0, -1,  0,  1,  0, ... ((-2/k) = A188510)
  1, -1,  0,  1, -1,  0,  1, -1,  0,  1, ... ((-3/k) = A102283)
  1,  0, -1,  0,  1,  0, -1,  0,  1,  0, ... ((-4/k) = A101455)
  1, -1,  1,  1,  0, -1,  1, -1,  1,  0, ... ((-5/k) = A226162)
  1,  0,  0,  0,  1,  0,  1,  0,  0,  0, ... ((-6/k) = A109017)
  1,  1, -1,  1, -1, -1,  0,  1,  1, -1, ... ((-7/k) = A175629)
  1,  0,  1,  0, -1,  0, -1,  0,  1,  0, ... ((-8/k) = A188510)
  ...

Eric Weisstein's World of Mathematics, Kronecker Symbol.

Cf. A215200.

The first rows are listed in A034947, A188510, A102283, A101455, A226162, A109017, A175629, A188510, ...

PARI
```
T(n,k) = kronecker(-n, k)
```

cp's OEIS Frontend

A103133 Decimal expansion of Dirichlet series L_{-7}(2).

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A322829 a(n) = Jacobi (or Kronecker) symbol (n/21).

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A126837 Ramanujan numbers (A000594) read mod 7^2.

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A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

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A091395 a(n) = Product_{ p | n } (1 + Legendre(-7,p) ).

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A323378 Square array read by antidiagonals: T(n,k) = Kronecker symbol (-n/k), n >= 1, k >= 1.

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