A038893 -id:A038893 - cp's OEIS frontend

A035203 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 21.

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

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 21. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

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.
Cf. A038893, A038894.

a[n_] := DivisorSum[n, KroneckerSymbol[21, #] &]; Array[a, 100] (* Amiram Eldar, Oct 11 2022 *)

my(m = 21); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(21, d)); \\ Amiram Eldar, Nov 19 2023

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log((5+sqrt(21))/2)/sqrt(21) = 0.683807... . - Amiram Eldar, Oct 11 2022
From Amiram Eldar, Nov 19 2023: (Start)
a(n) = Sum_{d|n} Kronecker(21, d).
Multiplicative with a(p^e) = 1 if Kronecker(21, p) = 0 (p = 3 or 7), a(p^e) = (1+(-1)^e)/2 if Kronecker(21, p) = -1 (p is in A038894), and a(p^e) = e+1 if Kronecker(21, p) = 1 (p is in A038893 \ {3, 7}). (End)

A082111 a(n) = n^2 + 5*n + 1.

Original entry on oeis.org

1, 7, 15, 25, 37, 51, 67, 85, 105, 127, 151, 177, 205, 235, 267, 301, 337, 375, 415, 457, 501, 547, 595, 645, 697, 751, 807, 865, 925, 987, 1051, 1117, 1185, 1255, 1327, 1401, 1477, 1555, 1635, 1717, 1801, 1887, 1975, 2065, 2157, 2251, 2347, 2445, 2545, 2647

Offset: 0

Paul Barry, Apr 04 2003

From Gary W. Adamson, Jul 29 2009: (Start)
Let (a,b) = roots to x^2 - 5*x + 1 = 0 = 4.79128... and 0.208712...
Then a(n) = (n + a) * (n + b). Example: a(5) = 51 = (5 + 4.79128...) * (5 + 0.208712...) (End)
For n > 0: a(n) = A176271(n+2,n). - Reinhard Zumkeller, Apr 13 2010
a(n-2) = n*(n+1) - 5, n >= 0, with a(-2) = -5 and a(-1) = -3, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 21 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013
Numbers m > 0 such that 4m+21 is a square. - Bruce J. Nicholson, Jul 19 2017
Numbers represented as 151 in number base B. If 'digits' from B upwards are allowed then 151(2)=15, 151(3)=25, 151(4)=37, 151(5)=51 also. - Ron Knott, Nov 14 2017
If A and B are sequences satisfying the recurrence t(n) = 5*t(n-1) - t(n-2) with initial values A(0) = 1, A(1) = n+5 and B(0) = -1, B(1) = n, then a(n) = A(i)^2 - A(i-1)*A(i+1) = B(j)^2 - B(j-1)*B(j+1) for i, j > 0. - Klaus Purath, Oct 18 2020
The prime terms in this sequence are listed in A089376. The prime factors are given in A038893. With the exception of 3 and 7, each prime factor p divides exactly 2 out of any p consecutive terms. If a(i) and a(k) form such a pair that are divisible by p, then i + k == -5 (mod p). - Klaus Purath, Nov 24 2020

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First row of A082110.

Cf. A002522, A014209, A028387, A028872, A338041.

Table[n^2 + 5*n + 1,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
LinearRecurrence[{3,-3,1},{1,7,15},80] (* Harvey P. Dale, Apr 22 2012 *)

a(n)=n^2+5*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n + a(n-1) + 4 (with a(0)=1). - Vincenzo Librandi, Aug 08 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=7, a(2)=15. - Harvey P. Dale, Apr 22 2012
Sum_{n>=0} 1/a(n) = 8/15 + Pi*tan(sqrt(21)*Pi/2)/sqrt(21) = 1.424563592286456286... . - Vaclav Kotesovec, Apr 10 2016
From G. C. Greubel, Jul 19 2017: (Start)
G.f.: (1 + 4*x - 3*x^2)/(1 - x)^3.
E.g.f.: (x^2 + 6*x + 1)*exp(x). (End)
a(n) = A014209(n+1) - 2 = A338041(2*n+1). - Hugo Pfoertner, Oct 08 2020
a(n) = A249547(n+1) - A024206(n-4), n >= 5. - Klaus Purath, Nov 24 2020

New title (using given formula) from Hugo Pfoertner, Oct 08 2020

A141159 Duplicate of A139492.

Original entry on oeis.org

7, 37, 43, 67, 79, 109, 127, 151, 163, 193, 211, 277, 331, 337, 373, 379, 421, 457, 463, 487, 499, 541, 547, 571, 613, 631, 673, 709, 739, 751, 757, 823, 877, 883, 907, 919, 967, 991, 1009, 1033, 1051, 1087, 1093, 1117, 1129, 1171, 1201, 1213, 1297, 1303

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (laucabfer(AT)alum.us.es), Jun 12 2008

dead

Name was: Primes of the form x^2 + 3*x*y - 3*y^2 (as well as of the form x^2 + 5*x*y + y^2).
Discriminant = 21. Class number = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2-4ac and gcd(a,b,c)=1 (primitive).
Primes of the form 6n+1 which cannot be expressed as 7k-1, 7k-2, or 7k-4. a(n)^2 == 1 (mod 24). - Gary Detlefs, Jan 26 2014
Besides 7 (which divides 21), primes of the form  p == 1 (mod 3) and either == 1 or 2 or 4 (mod 7). For the other class, the primes represented by the principal form [3, 3, -1] (or primitive forms equivalent to this) are besides 3 (which divides 21), congruent to 2 (mod 3) and also to 3, 5, 6 (mod 7). For the primes of both classes see A038893. - Wolfdieter Lang, Jun 19 2019

			a(1)=7 because we can write 7 = 2^2 + 3*2*1 - 3*1^2 (or 7 = 1^2 + 5*1*1 + 1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

f:=n->7*ceil((6*n+1)/7)-(6*n+1):for n from 1 to 220 do if isprime(6*n+1) and f(n)<>1 and f(n)<>2 and f(n)<>4 then print(6*n+1) fi od. # Gary Detlefs, Jan 26 2014

xy[{x_,y_}]:={x^2+3x y-3y^2,y^2+3x y -3x^2}; Union[Select[Flatten[xy/@ Subsets[ Range[50],{2}]],#>0&&PrimeQ[#]&]] (* Harvey P. Dale, Feb 17 2013 *)

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 3, -3])
Q.represented_positives(1326, 'prime') # Peter Luschny, Jun 24 2019

More terms from Harvey P. Dale, Feb 17 2013

cp's OEIS Frontend

A035203 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 21.

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A082111 a(n) = n^2 + 5*n + 1.

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A141159 Duplicate of A139492.

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cp's OEIS Frontend

A035203 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 21.

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PARI

Formula

A082111 a(n) = n^2 + 5*n + 1.

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Author

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Crossrefs

Programs

Mathematica

PARI

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A141159 Duplicate of A139492.

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Author

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Examples

References

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Programs

Maple

Mathematica

Sage

Extensions

A035203 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 21.