A033205 -id:A033205 - cp's OEIS frontend

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

T. D. Noe, May 09 2005, Apr 28 2008

Discriminant=-7. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac.

Consider sequences of primes produced by forms with -100

The Mathematica function QuadPrimes2 is useful for finding the primes less than "lim" represented by the positive definite quadratic form ax^2 + bxy + cy^2 for any a, b and c satisfying a>0, c>0, and discriminant d<0. It does this by examining all x>=0 and y>=0 in the ellipse ax^2 + bxy + cy^2 <= lim. To find the primes generated by positive and negative x and y, compute the union of QuadPrimes2[a,b,c,lim] and QuadPrimes2[a,-b,c,lim]. - T. D. Noe, Sep 01 2009

For other programs see the "Binary Quadratic Forms and OEIS" link.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Zak Seidov and N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (The first 1225 terms were found by Zak Seidov)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Discriminants in the range -3 to -100: A007645 (d=-3), A002313 (d=-4), A045373, A106856 (d=-7), A033203 (d=-8), A056874, A106857 (d=-11), A002476 (d=-12), A033212, A106858-A106861 (d=-15), A002144, A002313 (d=-16), A106862-A106863 (d=-19), A033205, A106864-A106865 (d=-20), A106866-A106869 (d=-23), A033199, A084865 (d=-24), A002476, A106870 (d=-27), A033207 (d=-28), A033221, A106871-A106874 (d=-31), A007519, A007520, A106875-A106876 (d=-32), A106877-A106881 (d=-35), A040117, A068228, A106882 (d=-36), A033227, A106883-A106888 (d=-39), A033201, A106889 (d=-40), A106890-A106891 (d=-43), A033209, A106282, A106892-A106893 (d=-44), A033232, A106894-A106900 (d=-47), A068229 (d=-48), A106901-A106904 (d=-51), A033210, A106905-A106906 (d=-52), A033235, A106907-A106913 (d=-55), A033211, A106914-A106917 (d=-56), A106918-A106922 (d=-59), A033212, A106859 (d=-60), A106923-A106930 (d=-63), A007521, A106931 (d=-64), A106932-A106933 (d=-67), A033213, A106934-A106938 (d=-68), A033246, A106939-A106948 (d=-71), A106949-A106950 (d=-72), A033212, A106951-A106952 (d=-75), A033214, A106953-A106955 (d=-76), A033251, A106956-A106962 (d=-79), A047650, A106963-A106965 (d=-80), A106966-A106970 (d=-83), A033215, A102271, A102273, A106971-A106974 (d=-84), A033256, A106975-A106983 (d=-87), A033216, A106984 (d=-88), A106985-A106989 (d=-91), A033217 (d=-92), A033206, A106990-A107001 (d=-95), A107002-A107008 (d=-96), A107009-A107013 (d=-99).
Other collections of quadratic forms: A139643, A139827.
For a more comprehensive list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. also A242660.

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 *)

list(lim)=my(q=Qfb(1,1,2), v=List([2])); forprime(p=2, lim, if(vecmin(qfbsolve(q, p))>0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 05 2016

Removed old Mathematica programs - T. D. Noe, Sep 09 2009

Edited (pointed out error in QuadPrimes, added new version of program, checked and extended b-file). - N. J. A. Sloane, Jun 06 2014

A140633 Primes of the form 7x^2+4xy+52y^2.

Original entry on oeis.org

7, 103, 127, 223, 367, 463, 487, 607, 727, 823, 967, 1063, 1087, 1303, 1327, 1423, 1447, 1543, 1567, 1663, 1783, 2143, 2287, 2383, 2503, 2647, 2767, 2887, 3343, 3463, 3583, 3607, 3727, 3823, 3847, 3943, 3967, 4327, 4423, 4447, 4567, 4663

Offset: 1

T. D. Noe, May 19 2008

Discriminant=-1440. Also primes of the forms 7x^2+6xy+87y^2 and 7x^2+2xy+103y^2.
Voight proves that there are exactly 69 equivalence classes of positive definite binary quadratic forms that represent almost the same primes. 48 of those quadratic forms are of the idoneal type discussed in A139827. The remaining 21 begin at A140613 and end here. The cross-references section lists the quadratic forms in the same order as tables 1-6 in Voight's paper. Note that A107169 and A139831 are in the same equivalence class.
In base 12, the sequence is 7, 87, X7, 167, 267, 327, 347, 427, 507, 587, 687, 747, 767, 907, 927, 9X7, X07, X87, XX7, E67, 1047, 12X7, 13X7, 1467, 1547, 1647, 1727, 1807, 1E27, 2007, 20X7, 2107, 21X7, 2267, 2287, 2347, 2367, 2607, 2687, 26X7, 2787, 2847, where X is for 10 and E is for 11. Moreover, the discriminant is X00 and that all elements are {7, 87, X7, 167, 187, 247} mod 260. - Walter Kehowski, May 31 2008

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
John Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.

Cf. A033205, A007519, A068228, A107135, A107144, A107152, A107151.
Cf. A107181, A139502, A139856, A139854, A139874, A139877, A139897.
Cf. A140613, A140614. A140615, A139923, A140616-A140619, A139988.
Cf. A139993, A140008, A140010, A140620-A140632, A033212, A102273.
Cf. A107007, A107003, A139830, A107169, A139831, A141373, A139847.
Cf. A139850, A139855, A139860, A139879, A139880, A139924, A139990.
Cf. A139998, A139992, A139996, A140003, A140013, A107006, A139858.
Cf. A107008, A140633, A139991, A139857, A139859, A139878, A007645, A002313.

Union[QuadPrimes2[7, 4, 52, 10000], QuadPrimes2[7, -4, 52, 10000]] (* see A106856 *)

A154778 Numbers of the form a^2 + 5b^2 with positive integers a,b.

Original entry on oeis.org

6, 9, 14, 21, 24, 29, 30, 36, 41, 45, 46, 49, 54, 56, 61, 69, 70, 81, 84, 86, 89, 94, 96, 101, 105, 109, 116, 120, 126, 129, 134, 141, 144, 145, 149, 150, 161, 164, 166, 174, 180, 181, 184, 189, 196, 201, 205, 206, 214, 216, 224, 225, 229, 230, 241, 244, 245, 246

Offset: 1

M. F. Hasler, Jan 24 2009

Subsequence of A020669 (which allows for a=0 and/or b=0). See there for further references. See A155560 ff for intersection of sequences of type (a^2 + k b^2).

Also, subsequence of A000408 (with 5b^2 = b^2 + (2b)^2).

			a(1) = 6 = 1^2 + 5*1^2 is the least number that can be written as A+5B where A,B are positive squares.
a(2) = 9 = 2^2 + 5*1^2 is the second smallest number that can be written in this way.

Cf. A033205 (subsequence of primes). [From R. J. Mathar, Jan 26 2009]

formQ[n_] := Reduce[a > 0 && b > 0 && n == a^2 + 5 b^2, {a, b}, Integers] =!= False; Select[ Range[300], formQ] (* Jean-François Alcover, Sep 20 2011 *)
Timing[mx = 300; limx = Sqrt[mx]; limy = Sqrt[mx/5]; Select[Union[Flatten[Table[x^2 + 5 y^2, {x, limx}, {y, limy}]]], # <= mx &]] (* T. D. Noe, Sep 20 2011 *)

isA154778(n,/* use optional 2nd arg to get other analogous sequences */c=5) = { for( b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))}
for( n=1,300, isA154778(n) & print1(n","))

A020669 Numbers of form x^2 + 5 y^2.

Original entry on oeis.org

0, 1, 4, 5, 6, 9, 14, 16, 20, 21, 24, 25, 29, 30, 36, 41, 45, 46, 49, 54, 56, 61, 64, 69, 70, 80, 81, 84, 86, 89, 94, 96, 100, 101, 105, 109, 116, 120, 121, 125, 126, 129, 134, 141, 144, 145, 149, 150, 161, 164, 166, 169, 174, 180, 181, 184, 189, 196, 201, 205, 206, 214, 216

Offset: 1

David W. Wilson

In other words, numbers represented by quadratic form with Gram matrix [1,0; 0,5].
x^2 + 5 y^2 has discriminant -20.
A positive integer n is in this sequence if and only if the p-adic order ord_p(n) of n is even for any prime p with floor(p/10) odd, and the number of prime divisors p == 3 or 7 (mod 20) of n with ord_p(n) odd has the same parity with ord_2(n). - Zhi-Wei Sun, Mar 24 2018

H. Cohn, A second course in number theory, John Wiley & Sons, Inc., New York-London, 1962. See pp. 3, 4 and later chapters.
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989. See Eq. (2.22), p. 33.

N. J. A. Sloane, Table of n, a(n) for n = 1..13859
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A033205, A106865, A154778, A216815, A122870.
For primes see A033205.
For the properly represented numbers see A344231.

[n: n in [0..216] | NormEquation(5, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

select(t -> [isolve(x^2+5*y^2=t)]<>[], [$0..1000]); # Robert Israel, May 11 2016

formQ[n_] := Reduce[x >= 0 && y >= 0 && n == x^2 + 5 y^2, {x, y}, Integers] =!= False; Select[ Range[0, 300], formQ] (* Jean-François Alcover, Sep 20 2011 *)
mx = 300;
limx = Sqrt[mx]; limy = Sqrt[mx/5];
Select[
Union[
Flatten[
Table[x^2 + 5*y^2, {x, 0, limx}, {y, 0, limy}]
       ]
     ], # <= mx &
] (* T. D. Noe, Sep 20 2011 *)

List contains 0 and all positive n such that 2*A035170(n) = A028586(2n) is nonzero. - Michael Somos, Oct 21 2006

Entry revised by N. J. A. Sloane, Sep 20 2012

A091727 Norms of prime ideals of Z[sqrt(-5)].

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 121, 127, 149, 163, 167, 169, 181, 223, 227, 229, 241, 263, 269, 281, 283, 289, 307, 347, 349, 361, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487

Offset: 1

Paul Boddington, Feb 02 2004

Consists of primes congruent to 1, 2, 3, 5, 7, 9 (mod 20) together with the squares of all other primes.
From Jianing Song, Feb 20 2021: (Start)
The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Note that Z[sqrt(-5)] has class number 2.
For primes p == 1, 9 (mod 20), there are two distinct ideals with norm p in Z[sqrt(-5)], namely (x + y*sqrt(-5)) and (x - y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p.
For p == 3, 7 (mod 20), there are also two distinct ideals with norm p, namely (p, x+y*sqrt(-5)) and (p, x-y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p^2 with y != 0; (2, 1+sqrt(-5)) and (sqrt(-5)) are respectively the unique ideal with norm 2 and 5.
For p == 11, 13, 17, 19 (mod 20), (p) is the only ideal with norm p^2. (End)

			From _Jianing Song_, Feb 20 2021: (Start)
Let |I| be the norm of an ideal I, then:
|(2, 1+sqrt(-5))| = 2;
|(3, 2+sqrt(-5))| = |(3, 2-sqrt(-5))| = 3;
|(sqrt(-5))| = 5;
|(7, 1+3*sqrt(-5))| = |(7, 1-3*sqrt(-5))| = 7;
|(23, 22+3*sqrt(-5))| = |(23, 22-3*sqrt(-5))| = 23;
|(3 + 2*sqrt(-5))| = |(3 - 2*sqrt(-5))| = 29;
|(6 + sqrt(-5))| = |(6 - sqrt(-5))| = 41. (End)

David A. Cox, Primes of the form x^2+ny^2, Wiley, 1989.
A. Frohlich and M. J. Taylor, Algebraic number theory, Cambridge university press, 1991.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A091728.
Cf. A289741, A033205, A106865, A139513, A003626.
The number of distinct ideals with norm n is given by A035170.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), this sequence (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA091727(n) = { my(ms = [1, 2, 3, 5, 7, 9], p, e=isprimepower(n,&p)); if(!e || e>2, 0, bitxor(e-1,!!vecsearch(ms,p%20))); }; \\ Antti Karttunen, Feb 24 2020

Offset corrected by Jianing Song, Feb 20 2021

A122487 2 together with odd primes p that divide Fibonacci[(p+1)/2].

Original entry on oeis.org

2, 13, 17, 37, 53, 73, 97, 113, 137, 157, 173, 193, 197, 233, 257, 277, 293, 313, 317, 337, 353, 373, 397, 433, 457, 557, 577, 593, 613, 617, 653, 673, 677, 733, 757, 773, 797, 853, 857, 877, 937, 953, 977, 997, 1013, 1033, 1093, 1097, 1117, 1153, 1193, 1213

Offset: 1

Alexander Adamchuk, Sep 16 2006

Primes of the form 2x^2+2xy+13y^2. Discriminant = -100. - T. D. Noe, May 02 2008

Primes of the form a^2 + b^2 such that a^2 == b^2 (mod 5). - Thomas Ordowski, May 18 2015

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A000045, A033205, A045468, A003631, A053028, A139827.

Select[Prime[Range[1000]],IntegerQ[Fibonacci[(#1+1)/2]/#1]&]

is(n)=my(k=n%20); (k==13||k==17||k==2) && isprime(n) \\ Charles R Greathouse IV, May 18 2015

Except for 2, the primes are congruent to {13, 17} (mod 20). - T. D. Noe, May 02 2008

2 together with all primes p == {13, 17} (mod 20). - Thomas Ordowski, May 18 2015

Definition changed by T. D. Noe, May 02 2008

A033718 Product theta3(q^d); d | 5.

Original entry on oeis.org

1, 2, 0, 0, 2, 2, 4, 0, 0, 6, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 8, 0, 0, 4, 2, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 4, 0, 0, 0, 6, 4, 0, 0, 6, 0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 8, 0, 4, 0, 0, 4, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 2, 0, 0, 0, 2, 12

Offset: 0

N. J. A. Sloane

nonn

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Number of representations of n as a sum of five times a square and a square. - Ralf Stephan, May 14 2007

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

Robert Israel, Table of n, a(n) for n = 0..10000
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2006-2007.
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. A000700, A000122, A010054, A020669, A033205, A121373.

S:= series(JacobiTheta3(0,q)*JacobiTheta3(0,q^5), q, 1001):
seq(coeff(S,q,j),j=0..1000); # Robert Israel, Dec 22 2015

terms = 127; s = EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^5] + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 04 2017 *)

{a(n)=if(n<1, n==0, qfrep([1,0;0,5],n)[n]*2)} /* Michael Somos, Aug 13 2006 */

N=666;  x='x+O('x^N);
T3(x)=1+2*sum(n=1,ceil(sqrt(N)),x^(n*n));
Vec(T3(x)*T3(x^5))
/* Joerg Arndt, Sep 21 2012 */

Theta series of lattice with Gram matrix [1 0 / 0 5].
Expansion of phi(q)phi(q^5) in powers of q where phi(q) is a Ramanujan theta function.
Euler transform of period 20 sequence [ 2, -3, 2, -1, 4, -3, 2, -1, 2, -6, 2, -1, 2, -3, 4, -1, 2, -3, 2, -2, ...]. - Michael Somos, Aug 13 2006
If p is prime then a(p) is nonzero iff p is in A033205.
0=a(n)a(2n) and 2*A035170(n) = a(n) + a(2n) if n>0. - Michael Somos, Oct 21 2006
a(n) is nonzero iff n is in A020669. - Robert Israel, Dec 22 2015
a(0) = 1, a(n) = (1+(-1)^t)b(n) for n > 0, where t is the number of prime factors of n, counting multiplicity, which are == 2,3,7 (mod 20), and b() is multiplicative with b(p^e) = (e+1) for primes p == 1,3,7,9 (mod 20) and b(p^e) = (1+(-1)^e)/2 for primes p == 11,13,17,19 (mod 20). (This formula is Corollary 3.3 in the Berkovich-Yesilyurt paper) - Jeremy Lovejoy, Nov 12 2024

A216815 Primes congruent to 1 or 9 mod 20.

Original entry on oeis.org

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, 1549, 1601, 1609, 1621, 1669, 1709, 1721, 1741, 1789, 1801, 1861

Offset: 1

N. J. A. Sloane, Sep 20 2012

This is a subsequence of A033205 but it is an important sequence in its own right.

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

A subsequence of A033205. Cf. A122870.

[p: p in PrimesUpTo(2000) | p mod 20 in [1, 9]]; // Vincenzo Librandi, Mar 22 2013

Select[Prime[Range[300]], MemberQ[{1, 9}, Mod[#, 20]]&] (* Vincenzo Librandi, Mar 22 2013 *)

Showing 1-10 of 23 results. Next

cp's OEIS Frontend

A140072 Values of x in A033205.

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A140073 Values of y in A033205.

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A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

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A140633 Primes of the form 7x^2+4xy+52y^2.

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A154778 Numbers of the form a^2 + 5b^2 with positive integers a,b.

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A020669 Numbers of form x^2 + 5 y^2.

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A091727 Norms of prime ideals of Z[sqrt(-5)].

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A122487 2 together with odd primes p that divide Fibonacci[(p+1)/2].

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