A306236 a(n) is the smallest integer m > n with integer j > m makes n^2, m^2 and j^2 an arithmetic progression.

5, 10, 15, 20, 25, 30, 13, 40, 45, 50, 55, 60, 65, 26, 75, 80, 25, 90, 95, 100, 39, 110, 37, 120, 125, 130, 135, 52, 145, 150, 41, 160, 165, 50, 65, 180, 185, 190, 195, 200, 85, 78, 215, 220, 225, 74, 65, 240, 61, 250, 75, 260, 265, 270, 275, 104, 285, 290

Offset

1,1

Comments

a(n) and n have the same parity.

If k is a term in A058529, gcd(k, a(k)) does not necessarily equal 1. For example, k = 217, 289, 343, 497, 529, 553, 679, 889, 961, 1127, ...

Conjecture: if gcd(k, a(k)) = 1, then k is a term in A058529.

Proof: if k is not in A058529, then k either is even or has a prime factor p == 3, 5 (mod 8). If k is even, then a(k) is also even, so 2 divides gcd(k, a(k)). If k has a prime factor p == 3, 5 (mod 8), then 2*m^2 == j^2 (mod p), 2^((p-1)/2)*m^(p-1) == -m^(p-1) == j^(p-1) (mod p), so m and j must both be multiples of p. As a result, p divides gcd(k, a(k)). - Jianing Song, Feb 09 2019

Links

Table of n, a(n) for n=1..58.

Formula

a(n) = sqrt((n^2 + A289398(n)^2)/2).

For positive integer k, a(2*k^2 - 1) = 2*k^2 + 2*k + 1.

a(A003629(k)) = 5*A003629(k).

a(n) <= 5*n.

a(k*n) = k*a(n) for all k not in A058529. - Jianing Song, Feb 15 2019

Example

a(1) = 5 because 1^2, 5^2 and 7^2 are an arithmetic progression.

Mathematica

Array[Block[{m = # + 2}, While[! IntegerQ@ Sqrt[2 m^2 - #^2], m += 2]; m] &, 58] (* Michael De Vlieger, Feb 15 2019 *)

Prog

(PARI) a(n) = {m=n+2; while(issquare(2*m^2-n^2)==0, m=m+2); m; }

Crossrefs

Cf. A003629, A058529, A289398 (integer j).

Sequence in context: A313730 A313731 A313732 * A297305 A182340 A313733

Adjacent sequences: A306233 A306234 A306235 * A306237 A306238 A306239

Keyword

nonn

Author

Jinyuan Wang, Feb 08 2019

Status

approved