A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.
2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821
Offset: 1
A000073 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = a(1) = 0 and a(2) = 1.
0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080, 98950096, 181997601, 334745777, 615693474, 1132436852
Offset: 0
Comments
The name "tribonacci number" is less well-defined than "Fibonacci number". The sequence A000073 (which begins 0, 0, 1) is probably the most important version, but the name has also been applied to A000213, A001590, and A081172. - N. J. A. Sloane, Jul 25 2024
Also (for n > 1) number of ordered trees with n+1 edges and having all leaves at level three. Example: a(4)=2 because we have two ordered trees with 5 edges and having all leaves at level three: (i) one edge emanating from the root, at the end of which two paths of length two are hanging and (ii) one path of length two emanating from the root, at the end of which three edges are hanging. - Emeric Deutsch, Jan 03 2004
a(n) is the number of compositions of n-2 with no part greater than 3. Example: a(5)=4 because we have 1+1+1 = 1+2 = 2+1 = 3. - Emeric Deutsch, Mar 10 2004
Let A denote the 3 X 3 matrix [0,0,1;1,1,1;0,1,0]. a(n) corresponds to both the (1,2) and (3,1) entries in A^n. - Paul Barry, Oct 15 2004
Number of permutations satisfying -k <= p(i)-i <= r, i=1..n-2, with k=1, r=2. - Vladimir Baltic, Jan 17 2005
Number of binary sequences of length n-3 that have no three consecutive 0's. Example: a(7)=13 because among the 16 binary sequences of length 4 only 0000, 0001 and 1000 have 3 consecutive 0's. - Emeric Deutsch, Apr 27 2006
Therefore, the complementary sequence to A050231 (n coin tosses with a run of three heads). a(n) = 2^(n-3) - A050231(n-3) - Toby Gottfried, Nov 21 2010
Convolved with the Padovan sequence = row sums of triangle A153462. - Gary W. Adamson, Dec 27 2008
For n > 1: row sums of the triangle in A157897. - Reinhard Zumkeller, Jun 25 2009
a(n+2) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 1; 0, 0, 1; 1, 0, 0] or [1, 1, 0; 1, 0, 1; 1, 0, 0] or [1, 1, 1; 1, 0, 0; 0, 1, 0] or [1, 0, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
a(n-1) is the top left entry of the n-th power of any of the 3 X 3 matrices [0, 0, 1; 1, 1, 1; 0, 1, 0], [0, 1, 0; 0, 1, 1; 1, 1, 0], [0, 0, 1; 1, 0, 1; 0, 1, 1] or [0, 1, 0; 0, 0, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
Least significant bits are given in A021913 (a(n) mod 2 = A021913(n)). - Andres Cicuttin, Apr 04 2016
The nonnegative powers of the tribonacci constant t = A058265 are t^n = a(n)*t^2 + (a(n-1) + a(n-2))*t + a(n-1)*1, for n >= 0, with a(-1) = 1 and a(-2) = -1. This follows from the recurrences derived from t^3 = t^2 + t + 1. See the example in A058265 for the first nonnegative powers. For the negative powers see A319200. - Wolfdieter Lang, Oct 23 2018
The term "tribonacci number" was coined by Mark Feinberg (1963), a 14-year-old student in the 9th grade of the Susquehanna Township Junior High School in Pennsylvania. He died in 1967 in a motorcycle accident. - Amiram Eldar, Apr 16 2021
Andrews, Just, and Simay (2021, 2022) remark that it has been suggested that this sequence is mentioned in Charles Darwin's Origin of Species as bearing the same relation to elephant populations as the Fibonacci numbers do to rabbit populations. - N. J. A. Sloane, Jul 12 2022
Examples
G.f. = x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 13*x^7 + 24*x^8 + 44*x^9 + 81*x^10 + ...
References
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Links
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Crossrefs
Cf. A000045, A000078, A000213, A000931, A001590 (first differences, also a(n)+a(n+1)), A001644, A008288 (tribonacci triangle), A008937 (partial sums), A021913, A027024, A027083, A027084, A046738 (Pisano periods), A050231, A054668, A062544, A063401, A077902, A081172, A089068, A118390, A145027, A153462, A230216.
Programs
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GAP
a:=[0,0,1];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Oct 24 2018
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Haskell
a000073 n = a000073_list !! n a000073_list = 0 : 0 : 1 : zipWith (+) a000073_list (tail (zipWith (+) a000073_list $ tail a000073_list)) -- Reinhard Zumkeller, Dec 12 2011 -
Magma
[n le 3 select Floor(n/3) else Self(n-1)+Self(n-2)+Self(n-3): n in [1..70]]; // Vincenzo Librandi, Jan 29 2016
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Maple
a:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^n)[1,3]: seq(a(n), n=0..40); # Alois P. Heinz, Dec 19 2016 # second Maple program: A000073:=proc(n) option remember; if n <= 1 then 0 elif n=2 then 1 else procname(n-1)+procname(n-2)+procname(n-3); fi; end; # N. J. A. Sloane, Aug 06 2018
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Mathematica
CoefficientList[Series[x^2/(1 - x - x^2 - x^3), {x, 0, 50}], x] a[0] = a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3]; Array[a, 36, 0] (* Robert G. Wilson v, Nov 07 2010 *) LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, May 24 2011 *) a[n_] := SeriesCoefficient[If[ n < 0, x/(1 + x + x^2 - x^3), x^2/(1 - x - x^2 - x^3)], {x, 0, Abs @ n}] (* Michael Somos, Jun 01 2013 *) Table[-RootSum[-1 - # - #^2 + #^3 &, -#^n - 9 #^(n + 1) + 4 #^(n + 2) &]/22, {n, 0, 20}] (* Eric W. Weisstein, Nov 09 2017 *) -
Maxima
A000073[0]:0$ A000073[1]:0$ A000073[2]:1$ A000073[n]:=A000073[n-1]+A000073[n-2]+A000073[n-3]$ makelist(A000073[n], n, 0, 40); /* Emanuele Munarini, Mar 01 2011 */
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PARI
{a(n) = polcoeff( if( n<0, x / ( 1 + x + x^2 - x^3), x^2 / ( 1 - x - x^2 - x^3) ) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, Sep 03 2007 */ -
PARI
my(x='x+O('x^99)); concat([0, 0], Vec(x^2/(1-x-x^2-x^3))) \\ Altug Alkan, Apr 04 2016 -
PARI
a(n)=([0,1,0;0,0,1;1,1,1]^n)[1,3] \\ Charles R Greathouse IV, Apr 18 2016, simplified by M. F. Hasler, Apr 18 2018
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Python
def a(n, adict={0:0, 1:0, 2:1}): if n in adict: return adict[n] adict[n]=a(n-1)+a(n-2)+a(n-3) return adict[n] # David Nacin, Mar 07 2012 from functools import cache @cache def A000073(n: int) -> int: if n <= 1: return 0 if n == 2: return 1 return A000073(n-1) + A000073(n-2) + A000073(n-3) # Peter Luschny, Nov 21 2022
Formula
G.f.: x^2/(1 - x - x^2 - x^3).
G.f.: x^2 / (1 - x / (1 - x / (1 + x^2 / (1 + x)))). - Michael Somos, May 12 2012
G.f.: Sum_{n >= 0} x^(n+2) *[ Product_{k = 1..n} (k + k*x + x^2)/(1 + k*x + k*x^2) ] = x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 13*x^7 + ... may be proved by the method of telescoping sums. - Peter Bala, Jan 04 2015
a(n) = central term in M^n * [1 0 0] where M = the 3 X 3 matrix [0 1 0 / 0 0 1 / 1 1 1]. (M^n * [1 0 0] = [a(n-1) a(n) a(n+1)].) a(n)/a(n-1) tends to the tribonacci constant, 1.839286755... = A058265, an eigenvalue of M and a root of x^3 - x^2 - x - 1 = 0. - Gary W. Adamson, Dec 17 2004
a(n+2) = Sum_{k=0..n} T(n-k, k), where T(n, k) = trinomial coefficients (A027907). - Paul Barry, Feb 15 2005
A001590(n) = a(n+1) - a(n); A001590(n) = a(n-1) + a(n-2) for n > 1; a(n) = (A000213(n+1) - A000213(n))/2; A000213(n-1) = a(n+2) - a(n) for n > 0. - Reinhard Zumkeller, May 22 2006
Let C = the tribonacci constant, 1.83928675...; then C^n = a(n)*(1/C) + a(n+1)*(1/C + 1/C^2) + a(n+2)*(1/C + 1/C^2 + 1/C^3). Example: C^4 = 11.444...= 2*(1/C) + 4*(1/C + 1/C^2) + 7*(1/C + 1/C^2 + 1/C^3). - Gary W. Adamson, Nov 05 2006
a(n) = j*C^n + k*r1^n + L*r2^n where C is the tribonacci constant (C = 1.8392867552...), real root of x^3-x^2-x-1=0, and r1 and r2 are the two other roots (which are complex), r1 = m+p*i and r2 = m-p*i, where i = sqrt(-1), m = (1-C)/2 (m = -0.4196433776...) and p = ((3*C-5)*(C+1)/4)^(1/2) = 0.6062907292..., and where j = 1/((C-m)^2 + p^2) = 0.1828035330..., k = a+b*i, and L = a-b*i, where a = -j/2 = -0.0914017665... and b = (C-m)/(2*p*((C-m)^2 + p^2)) = 0.3405465308... . - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
a(n+1) = 3*c*((1/3)*(a+b+1))^n/(c^2-2*c+4) where a=(19+3*sqrt(33))^(1/3), b=(19-3*sqrt(33))^(1/3), c=(586+102*sqrt(33))^(1/3). Round to the nearest integer. - Al Hakanson (hawkuu(AT)gmail.com), Feb 02 2009
a(n) = round(3*((a+b+1)/3)^n/(a^2+b^2+4)) where a=(19+3*sqrt(33))^(1/3), b=(19-3*sqrt(33))^(1/3).. - Anton Nikonov
Another form of the g.f.: f(z) = (z^2-z^3)/(1-2*z+z^4). Then we obtain a(n) as a sum: a(n) = Sum_{i=0..floor((n-2)/4)} ((-1)^i*binomial(n-2-3*i,i)*2^(n-2-4*i)) - Sum_{i=0..floor((n-3)/4)} ((-1)^i*binomial(n-3-3*i,i)*2^(n-3-4*i)) with natural convention: Sum_{i=m..n} alpha(i) = 0 for m > n. - Richard Choulet, Feb 22 2010
a(n+2) = Sum_{k=0..n} Sum_{i=k..n, mod(4*k-i,3)=0} binomial(k,(4*k-i)/3)*(-1)^((i-k)/3)*binomial(n-i+k-1,k-1). - Vladimir Kruchinin, Aug 18 2010
a(n) = 2*a(n-2) + 2*a(n-3) + a(n-4). - Gary Detlefs, Sep 13 2010
a(n) = 2*a(n-1) - a(n-4), with a(0)=a(1)=0, a(2)=a(3)=1. - Vincenzo Librandi, Dec 20 2010
Starting (1, 2, 4, 7, ...) is the INVERT transform of (1, 1, 1, 0, 0, 0, ...). - Gary W. Adamson, May 13 2013
G.f.: Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + x + x^2)/( x*(4*k+3 + x + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 09 2013
a(n+2) = Sum_{j=0..floor(n/2)} Sum_{k=0..j} binomial(n-2*j,k)*binomial(j,k)*2^k. - Tony Foster III, Sep 08 2017
Sum_{k=0..n} (n-k)*a(k) = (a(n+2) + a(n+1) - n - 1)/2. See A062544. - Yichen Wang, Aug 20 2020
From Yichen Wang, Aug 27 2020: (Start)
Sum_{k=0..n} a(k) = (a(n+2) + a(n) - 1)/2. See A008937.
Sum_{k=0..n} k*a(k) = ((n-1)*a(n+2) - a(n+1) + n*a(n) + 1)/2. See A337282. (End)
For n > 1, a(n) = b(n) where b(1) = 1 and then b(n) = Sum_{k=1..n-1} b(n-k)*A000931(k+2). - J. Conrad, Nov 24 2022
Conjecture: the congruence a(n*p^(k+1)) + a(n*p^k) + a(n*p^(k-1)) == 0 (mod p^k) holds for positive integers k and n and for all the primes p listed in A106282. - Peter Bala, Dec 28 2022
Sum_{k=0..n} k^2*a(k) = ((n^2-4*n+6)*a(n+1) - (2*n^2-2*n+5)*a(n) + (n^2-2*n+3)*a(n-1) - 3)/2. - Prabha Sivaramannair, Feb 10 2024
a(n) = Sum_{r root of x^3-x^2-x-1} r^n/(3*r^2-2*r-1). - Fabian Pereyra, Nov 23 2024
Extensions
Minor edits by M. F. Hasler, Apr 18 2018
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021
A106276 Number of distinct zeros of x^3-x^2-x-1 mod prime(n).
1, 0, 0, 1, 2, 1, 1, 1, 0, 1, 0, 0, 1, 1, 3, 3, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 3, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 3, 1, 1, 0, 0, 0, 1, 1, 3, 1, 0, 1, 0, 1, 1, 1, 0, 3, 1, 3, 1, 1, 1, 1, 1, 1, 3, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 3, 3, 1, 3, 3, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1
Offset: 1
Keywords
Comments
This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step recursions, A000073 and A001644. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is -44 = -4*11. The primes p yielding 3 distinct zeros, A106279, correspond to the periods of the sequences A000073(k) mod p and A001644(k) mod p having length less than p. The Lucas 3-step sequence mod p has two additional primes p for which the period is less than p: 2 and 11, which are factors of the discriminant -44. For p=11, the Fibonacci 3-step sequence mod p has a period of p(p-1).
Links
- J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.
- Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Crossrefs
Programs
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Mathematica
Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 150}]
A028952 Numbers represented by quadratic form with Gram matrix [ 3, 1; 1, 4 ].
0, 3, 4, 5, 9, 12, 15, 16, 20, 23, 25, 27, 31, 33, 36, 37, 44, 45, 48, 55, 59, 60, 64, 67, 69, 71, 75, 80, 81, 89, 92, 93, 97, 99, 100, 108, 111, 113, 115, 124, 125, 132, 135, 137, 141, 144, 147, 148, 155, 157, 159, 165, 176, 177, 179, 180, 181, 185, 188, 191, 192
Offset: 1
Keywords
Comments
Nonnegative integers of the form 3*x^2 + 2*x*y + 4*y^2, discriminant -44. - Ray Chandler, Jul 12 2014
Links
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Crossrefs
Primes: A106282.
Extensions
More terms from Larry Reeves (larryr(AT)acm.org), Mar 29 2000
A106307 Primes that yield a simple orbit structure in 3-step recursions.
3, 5, 23, 31, 37, 59, 67, 71, 89, 97, 103, 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, 587, 619, 631, 641, 643, 647, 653, 661, 691, 709, 719, 727, 751, 797, 823, 829
Offset: 1
Keywords
Comments
Consider the 3-step recursion x(k)=x(k-1)+x(k-2)+x(k-3) mod n. For any of the n^3 initial conditions x(1), x(2) and x(3) in Zn, the recursion has a finite period. When n is a prime in this sequence, all of the orbits, except the one containing (0,0,0), have the same length.
A prime p is in this sequence if either (1) the polynomial x^3-x^2-x-1 mod p has no zeros for x in [0,p-1] (see A106282) or (2) the polynomial has zeros, but none is a root of unity mod p. The first two primes in the second category are 103 and 587.
Links
- Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Crossrefs
Cf. A106285 (orbits of 3-step sequences).
A028953 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 4 ].
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
Keywords
Comments
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
Examples
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 + ...
Links
- 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
Programs
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Magma
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 */
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Mathematica
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 *) -
PARI
{a(n) = if( n<1, n==0, qfrep([3, 1; 1, 4], n)[n] * 2)}; /* Michael Somos, Jun 24 2011 */ -
PARI
{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 */
Formula
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
A107662 -n is the discriminant of cubic polynomials irreducible over Zp for primes p represented by only one binary quadratic form.
23, 31, 44, 59, 76, 83, 107, 108, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907
Offset: 1
Comments
Let f(x) be any monic integral cubic polynomial with discriminant -n and irreducible over Z. Consider the set S of primes p such that f(x) has no zeros in Zp, i.e., f(x) is irreducible in Zp. For the discriminants -n in this sequence, set S coincides with the primes represented by one binary quadratic form ax^2+bxy+cy^2 with -n=b^2-4ac. For examples, see A106867, A106872, A106282, A106919, A106954, A106967, A040034 and A040038. This sequence consists of (1) terms 4d in A106312 such that the class number of d is 1, (2) terms d in A106312 such that the class number of d is 3 and (3) 108 and 243.
Examples
For each -n, we give (-n,a,b,c) for the quadratic form ax^2+bxy+cy^2: (23,2,1,3), (31,2,1,4), (44,3,2,4), (59,3,1,5), (76,4,2,5), (83,3,1,7), (107,3,1,9), (108,4,2,7), (139,5,1,7), (172,4,2,11), (211,5,3,11), (243,7,3,9), (268,4,2,17), (283,7,5,11), (307,7,1,11), (331,5,3,17), (379,5,1,19), (499,5,1,25), (547,11,5,13), (643,7,1,23), (652,4,2,41), (883,13,1,17) and (907,13,9,19).
References
- Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
- Blair K. Spearman and Kenneth S. Williams, The cubic congruence x^3+Ax^2+Bx+C = 0 (mod p) and binary quadratic forms, J. London Math. Soc., 46, (1992), 397-410.
Links
- Eric Weisstein's World of Mathematics, Class Number
Comments
References
Links
Crossrefs
Programs
Mathematica
QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p] && p <= lmt && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]]; QuadPrimes2[1, 1, 2, 1000] (This is a corrected version of the old, incorrect, program QuadPrimes. - N. J. A. Sloane, Jun 15 2014) max = 1000; Table[yy = {y, 1, Floor[Sqrt[8 max - 7 x^2]/4 - x/4]}; Table[ x^2 + x y + 2 y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]& (* Jean-François Alcover, Oct 04 2018 *)PARI
Extensions