cp's OEIS Frontend

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

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

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)

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

This is one of the bisections of sequence A343238. The other sequence is A344232.

This is a proper subsequence of A020669.

The primes in this sequence are given in A033205.

Discriminant Disc = -20 = -4*5. Class number h(-20) = A000003(5) = 2. The reduced primitive forms representing the two proper (determinant = +1) equivalence classes are the present principal form F1 = [1, 0, 5] and F2 = [2, 2, 3] treated in A344232.

A positive integer k is properly represented by some primitive form of Disc = -20 if and only if the congruence s^2 + 20 == 0 (mod 4*k) has a solution. See, e.g., Buell Proposition 41, p. 50, or Scholz-Schoeneberg Satz 74, p. 105. That is, x^2 + 5 == 0 (mod k), with s = 2*x. For the representative solutions x from {0, 1, ..., k-1}, with k from A343238, see A343239. These solutions x determine the so-called representative parallel primitive forms (rpapfs) [k, 2*x, (x^2 + 5)/k] representing k. They are properly equivalent (via so called R(t)-transformations) to one of the reduced forms F1 or F2. (See also W. Lang's links in A225953 and A324251, but there indefinite forms are considered.)

In order to find out which k from A343238 is represented either by form F1 or F2 the two generic multiplicative characters of Disc = -20, namely Legendre(k|p), with the odd prime p = 5 which divides Disc = -20, and Jacobi(-1|k) can be used. See Buell, pp. 51-52. They lead to the two classes of genera of Disc -20.

The present genus I, the principal one, has for odd primes p, not 5, the values Legendre(p|5) = Legendre(5|p) = +1 and Jacobi(-1|p) = Legendre(-1|p) = +1, leading for odd primes not equal to 5 to A033205. The prime 2 is not represented. The prime 5 is trivially represented. For the other genus II these two characters have values -1. There prime 2 is represented.

For composite k the prime number factorization is used, and for powers of primes the lifting theorem is employed (see, e.g., Apostol, p. 121, Theorem 5.30). The solution for prime 2 represented by form F2 = [2, 2, 3] (from the other genus II) is not liftable to powers of 2. The solution for prime 5 is also not liftable (proof by induction). The solutions of the other primes from A033205 and A106865 are uniquely liftable to powers of these primes. See A343238 for all properly represented k for Disc = -20.

For the present genus I the properly represented integers k are given by 2^a*5^b*Product_{j=1..PI} (pI_j)^(eI(j))*Product_{k=1..PII} (pII_k)^(eII(k)), with a and b from {0, 1} but if PI = PII = 0 (empty products are 1) then a = b = 0 giving a(1) = 1. The odd primes pI_j are from A033205 (== {1, 9} (mod 20)), the primes pII_k are from the odd primes of A106865 (== {3, 7}(mod 20)). The exponents of the second product are restricted: if a = 1 then PII >= 1 and Sum_{k=1..PII} eII(k) is odd. If a = 0 then PII >= 0, and if PII >= 1 then this sum is even.

Neighboring numbers k (twins) begin: [5, 6], [29, 30], [45, 46], [69, 70], [205, 206], [229, 230], [245, 246], [269, 270], [405, 406], ...

For the solutions (X, Y) of F2 = [1, 0, 5] properly representing k = a(n) see A344233.

Ramanujan theta functions: f(q) := Prod_{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..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

The number of integer solutions (x, y) to 2*x^2 + 2*x*y + 3*y^2 = n, discriminant -20. - Ray Chandler, Jul 12 2014

See A106865 for all primes represented by this form.

Discriminant=-20.

Union of {2} and primes == 3 or 7 mod 20. - N. J. A. Sloane, Sep 04 2012

This is one of the bisections of sequence A343238. The other sequence is A344231.

This is a proper subsequence of A029718.

The primes in this sequence are given in A106865.

See A344231 for more details.

The reduced form [2, 2, 3] represents the proper (determinant +1) equivalence class of one of the two genera (genus II) of discriminant -20. The multiplicative generic characters for discriminant Disc = -20 have values Jacobi(a(n)|5) = -1 and Jacobi(-1|a(n)) = -1, for odd a(n) not divisible by 5. See Buell, p. 52.

The product of any two odd a(n), not divisible by 5, is congruent to {1,5} (mod 8). See Buell, 4), p. 51.

For this genus II of Disc = -20 the positive integers represented are given by 2^a*5^b*Product_{j=1..PI} (pI_j)^(eI(j))*Product_{k=1..PII}(pII_k)^(eII(k)), with a and b from {0, 1}, but if PI = PII = 0 (empty products are 1) then (a, b) = (1, 0) or (1, 1), giving a(1) = 2 or a(4) = 10. The odd primes pI_j are from A033205 and the odd primes pII_j from the odd primes of A106865. The exponents of the second product satisfy: if a = 1 then PII >= 0, and if PII >=1 then Sum_{k=1..PII} eII(j) is even. If a = 0 then PII >= 1 and this sum is odd.

The neighboring numbers k (twins) begin: [42, 43], [82, 83], [122, 123] [162, 163], [202, 203], [282, 283], ...

For the solutions (X, Y) of F2 = [2, 2, 3] properly representing k = a(n) see A344234.

Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 3 ].