A215653 -id:A215653 - cp's OEIS frontend

A076942 Smallest k > 0 such that n*k+1 is a square.

3, 4, 1, 2, 3, 4, 5, 1, 7, 8, 9, 2, 11, 12, 1, 3, 15, 16, 17, 4, 3, 20, 21, 1, 23, 24, 25, 6, 27, 4, 29, 7, 3, 32, 1, 8, 35, 36, 5, 2, 39, 4, 41, 10, 8, 44, 45, 1, 47, 48, 5, 12, 51, 52, 8, 3, 7, 56, 57, 2, 59, 60, 1, 15, 3, 8, 65, 16, 7, 12, 69, 4, 71, 72, 9, 18, 15, 8, 77, 1, 79, 80, 81

Offset: 1

Amarnath Murthy, Oct 19 2002

nonn

a(n) <= n-2 for n > 2; a(p) = p-2 if p is a prime > 2. [Comment corrected by Floris P. van Doorn, Jan 31 2009]

a(n) = n - 2 precisely when n > 2 has a primitive root; that is, for 4, and p^k and 2*p^k for p an odd prime and k > 0. - Franklin T. Adams-Watters, Apr 13 2009

Floris P. van Doorn, Table of n, a(n) for n = 1..10000
Dorin Andrica and Vlad Crişan, The Smallest Nontrivial Solution to x^k == 1 (mod n) and Related Sequences, Amer. Math. Monthly, Vol. 126, No. 2 (2019), pp. 173-178.

Cf. A033948, A033949, A215653. [From Franklin T. Adams-Watters, Apr 13 2009]

Do[k = 1; While[ !IntegerQ[Sqrt[n*k + 1]], k++ ]; Print[k], {n, 1, 85}]

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

a(n) = ((A215653(n))^2-1)/n.

Edited and extended by Robert G. Wilson v, Oct 21 2002

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

Original entry on oeis.org

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

Labos Elemer, Jun 08 2001

nonn

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.

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

Cf. A000005, A215653.

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

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

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

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

A215696 a(n)=smallest positive k>n+2 such that k*n+1 is a square.

Original entry on oeis.org

8, 12, 8, 12, 16, 20, 24, 15, 32, 36, 40, 24, 48, 52, 24, 33, 64, 68, 72, 42, 40, 84, 88, 35, 96, 100, 104, 60, 112, 56, 120, 69, 56, 132, 48, 78, 144, 148, 72, 60, 160, 72, 168, 96, 91, 180, 184, 63, 192, 196, 88, 114, 208, 212, 105, 85, 104, 228, 232, 84

Offset: 1

Zak Seidov, Aug 21 2012

nonn

For any n and k=n+2, 1+k*n=(n+1)^2, so here we consider the case k>n+2. Cases kA076942, A215653.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A076942, A215653.

for(n=1,100,k=n+3;while(!issquare(1+k*n),k++);print1(k","))

cp's OEIS Frontend

A076942 Smallest k > 0 such that n*k+1 is a square.

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

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

Formula

A215696 a(n)=smallest positive k>n+2 such that k*n+1 is a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI