A390047 - OEIS

A390047

Primes p that admit a quadratic orthomorphism f of Z_p such that f+c has cycle type (1,p-1) for all constants c.

23, 31, 41, 59, 71, 83, 89, 97, 103, 113, 131, 137, 139, 149, 151, 157, 163, 173, 179, 181, 197, 199, 211, 223, 227, 239, 251, 271, 283, 293, 307, 311, 331, 347, 349, 367, 379, 383, 389, 401, 409, 419, 431, 433, 461, 467, 487, 491, 499

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OFFSET

1,1

COMMENTS

Lists those primes p for which the following holds: there is a permutation f of Z_p that is of the form f(x)=a*x for x square, f(x)=b*x for x nonsquare, such that for every c the permutation x->f(x)+c has cycle type (1, p-1).

Such a permutation is in particular a quadratic orthomorphism of Z_p and a strong complete mapping of Z_p.

REFERENCES

A. B. Evans, Orthogonal Latin Squares Based on Groups, Vol. 57, Cham: Springer, 2018.

LINKS

Table of n, a(n) for n=1..49.

Paul Bastide, Anurag Bishnoi, Carla Groenland, Dion Gijswijt, and Rohinee Joshi, Circular sorting, strong complete mappings and wreath product constructions, arXiv:2510.18529 [math.CO], 2025.

EXAMPLE

For p=23 define f:Z_p->Z_p by

f(x)=5*x for x=0,1,2,3,4,6,8,9,12,13,16,18 (squares mod 23),

f(x)=7*x for x=5,7,10,11,14,15,17,19,20,21,22 (nonsquares mod 23).

CROSSREFS

Cf. A390046, A071607.

Sequence in context: A141818 A390613 A060328 * A034962 A133659 A309354

Adjacent sequences: A390044 A390045 A390046 * A390048 A390049 A390050

KEYWORD

nonn

AUTHOR

Dion Gijswijt, Oct 25 2025

STATUS

approved