A216815 -id:A216815 - cp's OEIS frontend

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

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

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

A122870 Primes congruent to 3 or 7 mod 20.

Original entry on oeis.org

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, 1223

Offset: 1

Alexander Adamchuk, Sep 16 2006

nonn

The old name was "Primes p that divide Lucas((p+1)/2) = A000032((p+1)/2)".

Note that F(p+1) = F((p+1)/2)*Lucas((p+1)/2), where F = A000045. Since gcd(F(n),Lucas(n)) = 1 or 2 (because Lucas(n)^2 - 5*F(n)^2 = 4*(-1)^n), this sequence (under the old definition above) lists primes p such that p divides F(p+1) but does not divides F((p+1)/2). By Propositions 1.1 and 1.2 (the k = 3 case) of my link below, this is primes p == 3, 7 (mod 20). - Jianing Song, Jun 20 2025

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
Jianing Song, Lucas sequences and entry point modulo p
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Gaussian Prime.

Cf. A000032, A000045, A122869, A216815.

Subseqeunce of A002145, A003631, A049098, A053027. Essentially the same as A106865.

[p: p in PrimesUpTo(1500) | p mod 20 in [3, 7]]; // Vincenzo Librandi, Jan 06 2013

Select[Prime[Range[1000]],IntegerQ[(Fibonacci[(#1+1)/2-1]+Fibonacci[(#1+1)/2+1])/#1]&]
Select[Prime[Range[300]], MemberQ[{3, 7}, Mod[#, 20]]&] (* Vincenzo Librandi, Jan 06 2013 *)

I merged A216816 into this entry at the suggestion of Jianing Song, Jun 20 2025. - N. J. A. Sloane, Jun 22 2025

A228141 Numbers that are congruent to {1, 5} mod 20.

Original entry on oeis.org

1, 5, 21, 25, 41, 45, 61, 65, 81, 85, 101, 105, 121, 125, 141, 145, 161, 165, 181, 185, 201, 205, 221, 225, 241, 245, 261, 265, 281, 285, 301, 305, 321, 325, 341, 345, 361, 365, 381, 385, 401, 405, 421, 425, 441, 445, 461, 465, 481, 485, 501, 505, 521, 525

Offset: 1

Colin Barker, Aug 12 2013

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A216815, A122870, A228142.

Flatten[Table[20n + {1, 5}, {n, 0, 24}]] (* Alonso del Arte, Aug 12 2013 *)

Vec(x*(15*x^2+4*x+1)/((x-1)^2*(x+1)) + O(x^100))

a(n) = -3*(4+(-1)^n) + 10*n.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: x*(15*x^2+4*x+1) / ((x-1)^2*(x+1)).
E.g.f.: 15 + (10*x - 12)*exp(x) - 3*exp(-x). - David Lovler, Sep 05 2022

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

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

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A122870 Primes congruent to 3 or 7 mod 20.

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Author

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Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A228141 Numbers that are congruent to {1, 5} mod 20.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula