cp's OEIS Frontend

The length of row n is r(n) = 2*A343240(b(n)), if A344231(n) = A343238(b(n)), for n >= 1. This sequence begins 2*(1, 1, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, ...).

The number of proper solutions (X, Y), with X > 0, is 1 for n = 1. X = 0 only for n = 2, and for n >= 2 only one half of the solutions are listed, namely those with X >= 0. There are also the solutions with (-X, -Y). Thus the total number of solutions for n >= 2 is actually r(n) given above.

For n >= 2 each distinct odd primes from {1, 3, 7, 9} (mod 20), i.e., from

A139513, that divides A344231(n) contributes a factor 2 to the number of solutions listed here. See A343238 and the corresponding A343240.

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

This sequence includes A139513, that is, Legendre(-5, p) = +1 for odd primes not 5, that is, primes congruent to {1, 3, 7, 9} mod 20. Here 5 is a member of the sequence with solution x = 0.

The primes of this sequence are given in A240920.

The present sequence gives the numbers of the form 2^a*5^b*Product_{j=1..m} (p_j)^e(j), with a and b from {0, 1}, p_j a prime from {1, 3, 7, 9} (mod 20), i.e., from A139513, m >= 0 and e(j) >= 0 (this includes the number 1). These numbers are ordered increasingly.

This follows from the Legendre-symbol(-5, p)= +1 and the lifting theorem (see, e.g., Apostol, Theorem 5.30, p. 121-2) for p = 2 and 5 (no lifting for the solutions for p = 2 and p = 5), and the unique lifting for the primes satisfying Legendre-symbol(-5, p) = +1.

Therefore the number of representative solutions x from {0, 1, ..., a(n)-1}, denoted by M(a(n)), is 1 for precisely four cases: a(1) = 1 (x = 0), a(2) = 2 (x = 1), a(4) = 5 (x = 0) and a(8) = 10 = 2*5 (x = 5). For each of the mentioned prime powers there are just 2 solutions. This implies that for the number of solutions in the general a(n) case, n not 1, 2, 4, 8, only the primes p_j are of interest: M(a(n)) = 2^m(n).

For these solutions x see A343239, and for the multiplicity M(a(n)) see A343240.

This congruence is needed to find all proper solutions of the positive definite binary quadratic form of discriminant Disc = -20 = -4*5 representing k = a(n). The solutions x lead to the so-called representative parallel primitive forms (rpapfs). See A344231 for more details.

For a bisection see A344231 and A344232, related to integer solutions of X^2 + 5*Y^2 = A344231(k) and 2*X^2 + 2*X*Y + 3*Y^2 = A344232(k).

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.

The length of row n is r(n) = 2*A343240(b(n)), if A344232(n) = A343238(b(n)), for n >= 1. This sequence begins 2*(1, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, ...).

See A344231 for references and links on parallel forms and half-reduced right neighbor forms (R-transformations), and also for the remark on the equivalent reduced form [2, -2, 3].

The number of proper solutions (X, Y), with X > 0, is 1 for n = 1 and 4. X = 0 only for n = 2, but the solution (0, -1) = (-0, -1) is not listed here.

For other n each distinct odd prime from {1, 3, 7, 9} (mod 20), i.e., from A139513, that divides A344232(n) contributes a factor of 2 to the listed number of solutions. See A343238 and A343240 for the multiplicities.

Only solutions with nonnegative X are listed. There is also the corresponding solution (-X, -Y). Hence the total number of signed solution is twice the number considered here.