A215653 a(n) = smallest positive m such that m^2 = 1+k*n with positive k.

2, 3, 2, 3, 4, 5, 6, 3, 8, 9, 10, 5, 12, 13, 4, 7, 16, 17, 18, 9, 8, 21, 22, 5, 24, 25, 26, 13, 28, 11, 30, 15, 10, 33, 6, 17, 36, 37, 14, 9, 40, 13, 42, 21, 19, 45, 46, 7, 48, 49, 16, 25, 52, 53, 21, 13, 20, 57, 58, 11, 60, 61, 8, 31, 14, 23, 66, 33, 22, 29

Offset: 1

Author

Zak Seidov, Aug 19 2012

Keywords

• nonn

Comments

Apparently a(n) = A070667(n) for n > 2. Note the linear patterns in the graph.

Examples

			a(1) = 2, k = 3; a(2) = 3, k = 4; a(3) = 2, k = 1; a(1000) = 249, k = 62.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)
• 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.

Crossrefs

Cf. A061369, A070667, A076942.

Programs

Mathematica

Flatten[{2,Table[Select[Range[2,1000],PowerMod[#,2,k]==1&,1],{k,2,1000}]}] (* first 1000 terms *)

PARI

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

Formula

a(n) = sqrt(1+n*A076942(n)).

a(n) = sqrt(A061369(n)). - Amiram Eldar, Mar 16 2025