A091729 -id:A091729 - cp's OEIS frontend

A055664 Norms of Eisenstein-Jacobi primes.

3, 4, 7, 13, 19, 25, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 121, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 289, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 529, 541, 547, 571

Offset: 1

N. J. A. Sloane, Jun 09 2000

These are the norms of the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.

Let us say that an integer n divides a lattice if there exists a sublattice of index n. Example: 3 divides the hexagonal lattice. Then A003136 (Loeschian numbers) is the sequence of divisors of the hexagonal lattice. Say that n is a "prime divisor" if the index-n sublattice is not contained in any other sublattice except the original lattice itself. The present sequence gives the prime divisors of the hexagonal lattice. Similarly, A055025 (Norms of Gaussian primes) is the sequence of "prime divisors" of the square lattice. - Jean-Christophe Hervé, Dec 04 2006

			There are 6 Eisenstein-Jacobi primes of norm 3, omega-omega^2 times one of the 6 units [ +-1, +-omega, +-omega^2 ] but only one up to equivalence.

R. K. Guy, Unsolved Problems in Number Theory, A16.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.

T. D. Noe, Table of n, a(n) for n = 1..1000
TheGrayCuber, Eisenstein Primes Visually, Youtube video.

Cf. A055665-A055668, A055025-A055029, A135461, A135462. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

The Z[sqrt(-5)] analogs are in A020669, A091727, A091728, A091729, A091730 and A091731.

Join[{3}, Select[Range[600], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) & ]] (* Jean-François Alcover, Oct 09 2012, from formula *)

is(n)=(isprime(n) && n%3<2) || (issquare(n,&n) && isprime(n) && n%3==2) \\ Charles R Greathouse IV, Apr 30 2013

Consists of 3; rational primes == 1 (mod 3) [A002476]; and squares of rational primes == -1 (mod 3) [A003627^2].

More terms from David Wasserman, Mar 21 2002

A033205 Primes of form x^2 + 5*y^2.

Original entry on oeis.org

5, 29, 41, 61, 89, 101, 109, 149, 181, 229, 241, 269, 281, 349, 389, 401, 409, 421, 449, 461, 509, 521, 541, 569, 601, 641, 661, 701, 709, 761, 769, 809, 821, 829, 881, 929, 941, 1009, 1021, 1049, 1061, 1069, 1109, 1129, 1181, 1201, 1229, 1249, 1289, 1301, 1321, 1361, 1381, 1409, 1429, 1481, 1489

Offset: 1

N. J. A. Sloane

It is a classical result that p is of the form x^2 + 5y^2 if and only if p = 5 or p == 1 or 9 mod 20 (see Cox, page 33). - N. J. A. Sloane, Sep 20 2012
Except for 5, also primes of the form x^2 + 25y^2. See A140633. - T. D. Noe, May 19 2008
Or, 5 and all primes p that divide Fibonacci((p - 1)/2) = A121568(n). - Alexander Adamchuk, Aug 07 2006

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 2000 terms from Vincenzo Librandi]
B. W. Brewer, On primes of the form u^2+5v^2, Am. Math. Monthly vol. 17 no 2 (1966) pp 502-509.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Subsequence of A091729.
Primes in A020669 (numbers of form x^2+5y^2). Cf. A121568, A139643, A216815.
Cf. A029718, A106865 (in the same genus).

[p: p in PrimesUpTo(2000) | NormEquation(5,p) eq true]; // Bruno Berselli, Jul 03 2016

QuadPrimes2[1, 0, 5, 10000] (* see A106856 *)

is(n)=my(k=n%20); n==5 || ((k==9 || k==9) && isprime(n)) \\ Charles R Greathouse IV, Feb 09 2017

A020669 INTERSECT A000040.

a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012

A119996 Numerator of Sum_{k=1..n} 1/(Fibonacci(k)*Fibonacci(k+2)).

Original entry on oeis.org

1, 5, 14, 39, 103, 272, 713, 1869, 4894, 12815, 33551, 87840, 229969, 602069, 1576238, 4126647, 10803703, 28284464, 74049689, 193864605, 507544126, 1328767775, 3478759199, 9107509824, 23843770273, 62423800997, 163427632718

Offset: 1

Alexander Adamchuk, Aug 03 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A000045, A059248, A064831, A001654, A045468, A064739, A091729.

F:=Fibonacci;; List([1..30], n-> F(n+1)*F(n+2)-1); # G. C. Greubel, Jul 23 2019

[Fibonacci(n+1)* Fibonacci(n+2)-1: n in [1..30]]; // Vincenzo Librandi, Aug 14 2012

with(combinat): seq(fibonacci(n+1)*fibonacci(n+2)-1, n=1..30); # Zerinvary Lajos, Jan 31 2008

Numerator[Table[Sum[1/(Fibonacci[k]*Fibonacci[k+2]),{k,n}],{n,30}]]
LinearRecurrence[{3,0,-3,1},{1,5,14,39},30] (* Harvey P. Dale, Aug 22 2011 *)

vector(30, n, f=fibonacci; f(n+1)*f(n+2)-1) \\ G. C. Greubel, Jul 23 2019

f=fibonacci; [f(n+1)*f(n+2)-1 for n in (1..30)] # G. C. Greubel, Jul 23 2019

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4); a(0)=1, a(1)=5, a(2)=14, a(3)=39. - Harvey P. Dale, Aug 22 2011
G.f.: ((x-2)*x-1)/(x^4 - 3*x^3 + 3*x - 1). - Harvey P. Dale, Aug 22 2011
a(n) = Fibonacci(n+1)*Fibonacci(n+2) - 1. - Gary Detlefs, Mar 31 2012
a(n) = Sum_{k=1..n} Fibonacci(k+1)^2. Can be proved by induction from Gary Detlefs formula. - Joel Courtheyn, Mar 15 2021

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A055664 Norms of Eisenstein-Jacobi primes.

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A033205 Primes of form x^2 + 5*y^2.

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A119996 Numerator of Sum_{k=1..n} 1/(Fibonacci(k)*Fibonacci(k+2)).

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