A028952 -id:A028952 - cp's OEIS frontend

A106282 Primes p such that the polynomial x^3-x^2-x-1 mod p has no zeros; i.e., the polynomial is irreducible over the integers mod p.

3, 5, 23, 31, 37, 59, 67, 71, 89, 97, 113, 137, 157, 179, 181, 191, 223, 229, 251, 313, 317, 331, 353, 367, 379, 383, 389, 433, 443, 449, 463, 467, 487, 509, 521, 577, 619, 631, 641, 643, 647, 653, 661, 691, 709, 719, 727, 751, 797, 823, 829, 839, 859, 881

Offset: 1

T. D. Noe, May 02 2005

nonn

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step sequences, A000073 and A001644.

Primes of the form 3x^2+2xy+4y^2 with x and y in Z. - T. D. Noe, May 08 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..300
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Primes in A028952.
Cf. A106276 (number of distinct zeros of x^3-x^2-x-1 mod prime(n)), A106294, A106302 (period of Lucas and Fibonacci 3-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible mod p).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 200}];Prime[Flatten[Position[t, 0]]]

forprime(p=2,1000,if(#polrootsmod(x^3-x^2-x-1,p)==0,print1(p,", ")));
/* Joerg Arndt, Jul 19 2012 */

A028953 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 4 ].

Original entry on oeis.org

1, 0, 0, 2, 2, 2, 0, 0, 0, 2, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 4, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 6, 2, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 8, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 2, 0, 0, 0, 2, 0, 2, 6, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The number of integer solutions (x, y) to 3*x^2 + 2*x*y + 4*y^2, discriminant -44. - Ray Chandler, Jul 12 2014

			G.f. = 1 + 2*q^3 + 2*q^4 + 2*q^5 + 2*q^9 + 4*q^12 + 2*q^15 + 2*q^16 + 4*q^20 + 2*q^23 + 2*q^25 + 2*q^27 + 2*q^31 + 2*q^33 + 6*q^36 + 2*q^37 + 2*q^44 + 4*q^45 + ...

John Cannon, Table of n, a(n) for n = 0..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A028952, A106282.

A := Basis( ModularForms( Gamma1(44), 1), 87); A[1] + 2*A[4] + 2*A[5] + 2*A[6] + 2*A[10] + 4*A[13] + 2*A[16] + 2*A[17] + 4*A[21] + 2*A[24]; /* Michael Somos, Feb 09 2017 */

r[n_] := Reduce[{x, y}.{{3, 1}, {1, 4}}.{x, y} == n, {x, y}, Integers]; Table[rn = r[n]; Which[rn === False, 0, Head[rn] === Or, Length[rn], Head[rn] === And, 1], {n, 0, 105}] (* Jean-François Alcover, Nov 05 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^11] - 2 q QPochhammer[ q^2] QPochhammer[ q^22], {q, 0, n}]; (* Michael Somos, Feb 09 2017 *)

{a(n) = if( n<1, n==0, qfrep([3, 1; 1, 4], n)[n] * 2)}; /* Michael Somos, Jun 24 2011 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=1, sqrtint( n), 2 * x^k^2, 1 + A) * sum( k=1, sqrtint( n\11), 2 * x^(11*k^2), 1 + A) - 2 * x * eta(x^2 + A) * eta(x^22 + A), n))}; /* Michael Somos, Jun 24 2011 */

Expansion of phi(q) * phi(q^11) - 2*q * f(-q^2) * f(-q^22) = phi(q^3) * phi(q^33) + 2*q^4 * chi(q) * psi(-q^3) * chi(q^11) * psi(-q^33) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos and Alex Berkovich, Jun 24 2011
G.f. is a period 1 Fourier series which satisfies f(-1 / (44 t)) = 44^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 24 2011
G.f.: Sum_{n, m in Z} x ^ (3*n*n + 2*n*m + 4*m*m).
a(4*n + 2) = a(11*n + 2) = a(11*n + 6) = a(11*n + 7) = a(11*n + 8) = a(11*n + 10) = 0. - Michael Somos, Feb 23 2012

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A106282 Primes p such that the polynomial x^3-x^2-x-1 mod p has no zeros; i.e., the polynomial is irreducible over the integers mod p.

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A028953 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 4 ].

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