A020669 -id:A020669 - cp's OEIS frontend

A155565 Intersection of A001481 and A020669: N = a^2 + b^2 = c^2 + 5d^2 for some integers a,b,c,d.

0, 1, 4, 5, 9, 16, 20, 25, 29, 36, 41, 45, 49, 61, 64, 80, 81, 89, 100, 101, 109, 116, 121, 125, 144, 145, 149, 164, 169, 180, 181, 196, 205, 225, 229, 241, 244, 245, 256, 261, 269, 281, 289, 305, 320, 324, 349, 356, 361, 369, 389, 400, 401, 404, 405, 409, 421

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155575 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155565(n,/* use optional 2nd arg to get other analogous sequences */c=[5,1]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,500, isA155565(n) & print1(n","))

A155567 Intersection of A002479 and A020669 : N = a^2 + 2b^2 = c^2 + 5d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 6, 9, 16, 24, 25, 36, 41, 49, 54, 64, 81, 86, 89, 96, 100, 121, 129, 134, 144, 150, 164, 166, 169, 196, 201, 214, 216, 225, 241, 246, 249, 256, 281, 289, 294, 321, 324, 326, 344, 356, 361, 369, 384, 400, 401, 409, 441, 449, 454, 484, 486, 489, 516, 521

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155577 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155567(n,/* use optional 2nd arg to get other analogous sequences */c=[5,2]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,600, isA155567(n) & print1(n","))

A155570 Intersection of A003136 and A020669: N = a^2 + 3b^2 = c^2 + 5d^2 for some integers a,b,c,d.

Original entry on oeis.org

0, 1, 4, 9, 16, 21, 25, 36, 49, 61, 64, 81, 84, 100, 109, 121, 129, 144, 169, 181, 189, 196, 201, 225, 229, 241, 244, 256, 289, 301, 309, 324, 336, 349, 361, 381, 400, 409, 421, 436, 441, 469, 484, 489, 516, 525, 529, 541, 549, 576, 601, 625, 661, 669, 676, 709

Offset: 1

M. F. Hasler, Jan 25 2009

Contains A155710 as a subsequence (obtained by restricting a,b,c,d to be nonzero). Also contains A000290 (squares) as subsequence.

Cf. A001481, A002479, A003136, A002481, A020668 ff.

isA155570(n,/* use optional 2nd arg to get other analogous sequences */c=[5,3]) = { for(i=1,#c, for(b=0,sqrtint(n\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=0,800, isA155570(n) & print1(n","))

A302982 Number of ways to write n as x^2 + 5y^2 + 2^z + 32^w with x,y,z,w nonnegative integers.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 2, 4, 3, 3, 5, 4, 6, 7, 4, 7, 5, 4, 7, 8, 5, 5, 8, 5, 9, 7, 6, 13, 10, 7, 9, 10, 7, 12, 11, 8, 11, 7, 7, 11, 11, 6, 11, 13, 6, 10, 7, 7, 17, 13, 6, 13, 14, 9, 11, 18, 10, 13, 14, 11

Offset: 1

Zhi-Wei Sun, Apr 16 2018

nonn

Conjecture: a(n) > 0 for all n > 3.
Clearly, a(4*n) > 0 if a(n) > 0. We have verified a(n) > 0 for all n = 4..2*10^8.
See also A302983 and A302984 for similar conjectures.

			a(4) = 1 with 4 = 0^2 + 5*0^2 + 2^0 + 3*2^0.
a(5) = 2 with 5 =  1^2 + 5*0^2 + 2^0 + 3*2^0 = 0^2 + 5*0^2 + 2^1 + 3*2^0.
a(6) = 1 with 6 = 1^2 + 3*0^2 + 2^1 + 3*2^0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000079, A000290, A020669, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A301534, A302920, A302981, A302983, A302984.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-3*2^k-2^j-5x^2],r=r+1],{k,0,Log[2,n/3]},{j,0,If[n==3*2^k,-1,Log[2,n-3*2^k]]},{x,0,Sqrt[(n-3*2^k-2^j)/5]}];tab=Append[tab,r],{n,1,60}];Print[tab]

A055664 Norms of Eisenstein-Jacobi primes.

Original entry on oeis.org

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

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","))

A002481 Numbers of form x^2 + 6y^2.

Original entry on oeis.org

0, 1, 4, 6, 7, 9, 10, 15, 16, 22, 24, 25, 28, 31, 33, 36, 40, 42, 49, 54, 55, 58, 60, 63, 64, 70, 73, 79, 81, 87, 88, 90, 96, 97, 100, 103, 105, 106, 112, 118, 121, 124, 127, 132, 135, 144, 145, 150, 151, 154, 159, 160, 166, 168, 169, 175, 177, 186, 193, 196, 198, 199, 202, 214

Offset: 1

N. J. A. Sloane

nonn

Norms of numbers in Z[sqrt(-6)]. - Alonso del Arte, Sep 23 2014

It seems that 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/12) odd, and the number of prime divisors p == 5 or 11 (mod 24) with ord_p(n) odd has the same parity with ord_2(n) + ord_3(n). - Zhi-Wei Sun, Mar 24 2018

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..2064 (terms <= 10000).
Leonhard Euler, E388 Vollständige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 425.

Cf. A020669, A033199.

N:= 10^4: # to get all terms <= N
{seq(seq(a^2 + 6*b^2, a = 0 .. floor(sqrt(N-6*b^2))), b = 0 .. floor(sqrt(N/6)))};
# for Maple 11, or earlier, uncomment the next line
# sort(convert(%,list));  # Robert Israel, Sep 24 2014

lim = 10^4; k = 6; Union@Flatten@Table[x^2 + k * y^2, {y, 0, Sqrt[lim/k]}, {x, 0, Sqrt[lim - k * y^2]}] (* Zak Seidov, Mar 30 2011 *)

A106865 Primes of the form 2x^2 + 2xy + 3y^2.

Original entry on oeis.org

2, 3, 7, 23, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 263, 283, 307, 347, 367, 383, 443, 463, 467, 487, 503, 523, 547, 563, 587, 607, 643, 647, 683, 727, 743, 787, 823, 827, 863, 883, 887, 907, 947, 967, 983, 1063, 1087, 1103, 1123, 1163, 1187

Offset: 1

T. D. Noe, May 09 2005

Discriminant = -20.
Also: Primes of the form 2x^2 - 2xy + 3y^2 with x and y nonnegative. Cf. A106864.
Primes congruent to 2, 3, 7 modulo 20. - Michael Somos, Aug 13 2006
In Z[sqrt(-5)], these numbers are irreducible but not prime. In terms of ideals, they generate principal ideals that are not prime (or maximal). The equation x^2 + 5y^2 = a(n) has no solutions, but x^2 = -5 (mod a(n)) does. For example, 2 * 3 = (1 - sqrt(-5))(1 + sqrt(-5)) and 7 * 23 = (9 - 4*sqrt(-5))(9 + 4*sqrt(-5)). - Alonso del Arte, Dec 19 2015

			x = 1, y = 1 gives 2x^2 + 2xy + 3y^2 = 2 + 2 + 3 = 7.
x = 1, y = -3 gives 2x^2 + 2xy + 3y^2 = 2 - 6 + 27 = 23.

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 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

For n > 1, a(n) = A122870(n-1). Cf. A122870, A106864.

select(isprime, [2, seq(seq(5+s+20*i,s=[-2,2]),i=0..10^3)]); # Robert Israel, Dec 23 2015

QuadPrimes2[2, -2, 3, 10000] (* see A106856 *)

is(n)=isprime(n) && #qfbsolve(Qfb(2,2,3),n)>0 \\ Charles R Greathouse IV, Feb 09 2017

Complement(A000040, A020669).

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

cp's OEIS Frontend

A155565 Intersection of A001481 and A020669: N = a^2 + b^2 = c^2 + 5d^2 for some integers a,b,c,d.

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A155567 Intersection of A002479 and A020669 : N = a^2 + 2b^2 = c^2 + 5d^2 for some integers a,b,c,d.

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A155570 Intersection of A003136 and A020669: N = a^2 + 3b^2 = c^2 + 5d^2 for some integers a,b,c,d.

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A302982 Number of ways to write n as x^2 + 5*y^2 + 2^z + 3*2^w with x,y,z,w nonnegative integers.

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

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A002481 Numbers of form x^2 + 6y^2.

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A302982 Number of ways to write n as x^2 + 5y^2 + 2^z + 32^w with x,y,z,w nonnegative integers.