A061369 a(n) = smallest square in the arithmetic progression {n*k+1 : k > 0}.

4, 9, 4, 9, 16, 25, 36, 9, 64, 81, 100, 25, 144, 169, 16, 49, 256, 289, 324, 81, 64, 441, 484, 25, 576, 625, 676, 169, 784, 121, 900, 225, 100, 1089, 36, 289, 1296, 1369, 196, 81, 1600, 169, 1764, 441, 361, 2025, 2116, 49, 2304, 2401, 256, 625, 2704, 2809, 441

Offset: 1

Author

Labos Elemer, Jun 08 2001

Keywords

• nonn

Comments

Each square q appears in several progressions, i.e., in A000005(q-1) cases. Not necessarily as a first square. E.g., q = 81 is minimal in {10*k+1}, {20*k+1}, {40*k+1}, {80*k+1}, while not minimal in {3*k+1} and 5 other progressions. In {16*k+1} 49 comes before 81.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A215653.

Programs

Mathematica

a[n_] := Module[{m = n + 1}, While[!IntegerQ[Sqrt[m]], m += n]; m]; Array[a, 55] (* Amiram Eldar, Mar 16 2025 *)

PARI

a(n) = {my(m = n + 1); while(!issquare(m), m += n); m;} \\ Amiram Eldar, Mar 16 2025

Formula

a(n) = Min{x^2 : x^2 = n*k+1}.

a(n) = A215653(n)^2. - Amiram Eldar, Mar 16 2025