A020673 -id:A020673 - cp's OEIS frontend

A033201 Primes of the form x^2 + 10*y^2.

11, 19, 41, 59, 89, 131, 139, 179, 211, 241, 251, 281, 331, 379, 401, 409, 419, 449, 491, 499, 521, 569, 571, 601, 619, 641, 659, 691, 739, 761, 769, 809, 811, 859, 881, 929, 971, 1009, 1019, 1049, 1051, 1091, 1129, 1171, 1201, 1249, 1259, 1289, 1291, 1321, 1361, 1409, 1451, 1459, 1481

Offset: 1

N. J. A. Sloane

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 36.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 2000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643.

Primes in A020673.

[p: p in PrimesUpTo(1500) | NormEquation(10,p) eq true]; // Bruno Berselli, Jul 03 2016

Clear[f,lst,p,x,y]; f[x_,y_]:=x^2+10*y^2; lst={};Do[Do[p=f[x,y];If[PrimeQ[p]&&p<7212,AppendTo[lst,p]],{y,0,6!}],{x,0,6!}];Take[Union[lst],222] (* Vladimir Joseph Stephan Orlovsky, Aug 04 2009 *)
QuadPrimes2[1, 0, 10, 10000] (* see A106856 *)

select(n->vecsearch([1,9,11,19],n%40), primes(100)) \\ Charles R Greathouse IV, Nov 09 2012

Same as primes congruent to 1, 9, 11, or 19 mod 40. See, e.g., Cox, p. 36.

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

A216579 Number of positive integer solutions to the equation a^2 + 10*b^2 = n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2

Offset: 1

V. Raman, Sep 08 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A020673, A033201.

a[ n_] := If[ n < 1, 0, Length@Select[ Range@Sqrt[ n/10], IntegerQ@Sqrt[ n - 10 #] &]]; (* Michael Somos, Jan 10 2015 *)

a(n)=sum(b=1,sqrtint((n-1)\10),issquare(n-10*b^2)) \\ Charles R Greathouse IV, Nov 19 2014

A216577 Number of nonnegative integer solutions to the equation x^2 + 10*y^2 = n.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 0

V. Raman, Sep 08 2012

nonn

Cf. A020673, A033201.

Corrected by T. D. Noe and N. J. A. Sloane, Sep 10 2012

A301805 Number of ways to write 3n^2 as x^2 + 10y^2 + 2^z, where x, y and z are nonnegative integers with z > 3.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 4, 4, 4, 4, 4, 4, 6, 4, 4, 3, 5, 4, 4, 5, 7, 5, 4, 4, 6, 4, 7, 5, 5, 7, 7, 5, 5, 4, 8, 5, 7, 6, 11, 6, 6, 5, 8, 5, 6, 7, 5, 7, 6, 5, 5, 5, 7, 7, 4, 4, 8, 8, 8, 6, 6, 6, 9, 8, 8, 7, 8, 6, 10, 6, 10, 6, 8, 8, 8, 5

Offset: 1

Zhi-Wei Sun, Mar 27 2018

nonn

It might seem that a(n) > 0 for all n > 2. However, we find that a(323525083) = 0, moreover 3*323525083^2 cannot be written as x^2 + 10*y^2 + 2^z with x,y,z nonnegative integers. We note also that a(270035155) = 0 but 3*270035155^2 - 2^0 has the form x^2 + 10*y^2 with x and y integers.
My way to check whether 3*n^2 can be written as x^2 + 10*y^2 + 2^z is to find z such that 3*n^2 - 2^z can be written as x^2 + 10*y^2. I observe that a positive integer n has the form x^2 + 10*y^2 with x and y integers if and only if the p-adic order ord_p(n) of n is even for any prime p == 3, 17, 21, 27, 29, 31, 33, 39 (mod 40) and the sum of those ord_p(n) with p prime and p == 2, 5, 7, 13, 23, 37 (mod 40) is even.
From David A. Corneth, Mar 27 2018: (Start)
If a(n) > 0 then a(2*n) > 0; 3*n^2 = x^2 + 10*y^2 + 2^z <=> 3*(2*n)^2 = 4 * 3*n^2 = 4 * (x^2 + 10*y^2 + 2^z) = (2*x)^2 + 10 * (2*y)^2 + 2^(z + 2).
So we just need to check odd n and as z > 0, 2 | 2^z and furthermore 2 | 10 * y^2 so 3*x^2 must be odd, i.e., x must be odd for 3*n^2 to be odd. Also, y must be odd. For odd n, 3*n^2 == 3 (mod 4), for odd x, x^2 == (1 mod 4), for z >= 3, 2^z == 0 (mod 4) so 10 * y^2 must be == 2 (mod 4) which happens if and only if y is odd. (End)

			a(1) = a(2) = 0 since 3*1^2 < 3*2^2 < 2^4.
a(3) = 1 since 3*3^2 = 1^2 + 10*1^2 + 2^4.
a(4) = 1 since 3*4^2 = 4^2 + 10*0^2 + 2^5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..800
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000079, A000290, A020673, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301472, A301479, A301579, A301640.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[3*n^2-2^k-10x^2],r=r+1],{k,4,Log[2,3n^2]},{x,0,(3*n^2-2^k)/10}];tab=Append[tab,r],{n,1,80}];Print[tab]

A317641 Expansion of theta_3(q)*theta_3(q^10), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 02 2018

nonn

Number of integer solutions to the equation x^2 + 10*y^2 = n.

			G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^10 + 4*q^11 + 4*q^14 + 2*q^16 + 4*q^19 + ...

Seiichi Manyama, 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)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A000024, A020673, A033201, A033723, A216577, A216579, A258034.

nmax = 100; CoefficientList[Series[EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^10], {q, 0, nmax}], q]
nmax = 100; CoefficientList[Series[QPochhammer[-q, -q] QPochhammer[-q^10, -q^10]/(QPochhammer[q, -q] QPochhammer[q^10, -q^10]), {q, 0, nmax}], q]

G.f.: Product_{k>=1} (1 + x^(2*k-1))^2*(1 - x^(2*k))*(1 + x^(20*k-10))^2*(1 - x^(20*k)).

A216578 Prime numbers which cannot be written in the form x^2 + 10*y^2.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 113, 127, 137, 149, 151, 157, 163, 167, 173, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 257, 263, 269, 271, 277, 283, 293, 307, 311, 313, 317, 337, 347

Offset: 1

V. Raman, Sep 08 2012

nonn

Prime numbers not congruent to {1, 9, 11, 19} mod 40.

Cf. A020673, A033201.

A216580 Numbers which can be written in the form x^2 + 10*y^2, where x > 0 and y > 0.

Original entry on oeis.org

11, 14, 19, 26, 35, 41, 44, 46, 49, 56, 59, 65, 74, 76, 89, 91, 94, 99, 104, 106, 110, 115, 121, 126, 131, 139, 140, 154, 161, 164, 169, 171, 176, 179, 184, 185, 190, 196, 206, 209, 211, 224, 234, 235, 236, 241, 251, 254, 259, 260, 265, 266, 275, 281, 286

Offset: 1

V. Raman, Sep 08 2012

nonn

Cf. A020673, A033201.

cp's OEIS Frontend

A033201 Primes of the form x^2 + 10*y^2.

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A216579 Number of positive integer solutions to the equation a^2 + 10*b^2 = n.

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A216577 Number of nonnegative integer solutions to the equation x^2 + 10*y^2 = n.

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A301805 Number of ways to write 3*n^2 as x^2 + 10*y^2 + 2^z, where x, y and z are nonnegative integers with z > 3.

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A317641 Expansion of theta_3(q)*theta_3(q^10), where theta_3() is the Jacobi theta function.

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A216578 Prime numbers which cannot be written in the form x^2 + 10*y^2.

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A216580 Numbers which can be written in the form x^2 + 10*y^2, where x > 0 and y > 0.

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A301805 Number of ways to write 3n^2 as x^2 + 10y^2 + 2^z, where x, y and z are nonnegative integers with z > 3.