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

Author

Amarnath Murthy, Oct 19 2002

Keywords

• nonn

Comments

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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

Extensions

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