A257231 a(n) = n^2 mod p where p is the least prime greater than n.

1, 1, 4, 1, 4, 1, 5, 9, 4, 1, 4, 1, 16, 9, 4, 1, 4, 1, 16, 9, 4, 1, 7, 25, 16, 9, 4, 1, 4, 1, 36, 25, 16, 9, 4, 1, 16, 9, 4, 1, 4, 1, 16, 9, 4, 1, 36, 25, 16, 9, 4, 1, 36, 25, 16, 9, 4, 1, 4, 1, 36, 25, 16, 9, 4, 1, 16, 9, 4, 1, 4, 1, 36, 25, 16, 9, 4, 1, 16, 9, 4, 1, 36, 25, 16, 9, 4

Offset: 1

Author

Chris Boyd, Apr 19 2015

Keywords

• nonn
• easy

Comments

Conjecture: a(n) is always a positive square, except for the terms 5, 7, 69, 42 and 17 given by n = 7, 23, 113, 114, and 115 respectively. It is easy to show that nonsquare terms are in [p, q) iff p and q are consecutive primes and q-p > sqrt(q). There are no gaps between consecutive primes greater than sqrt(q) for 127 < q < 4*10^18 (see Nicely's table of maximal prime gaps).

Examples

			a(23) = 7 because 23^2 mod 29 = 7.
a(24) = 25 because 24^2 mod 29 = 25.
		

Links

• Chris Boyd, Table of n, a(n) for n = 1..10000
• Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
• Tomás Oliveira e Silva, Gaps between consecutive primes

Crossrefs

Cf. A257230.

Programs

Magma

[n^2 mod NextPrime(n): n in [1..80]]; // Vincenzo Librandi, Apr 19 2015

Mathematica

Table[Mod[n^2, NextPrime@ n], {n, 87}] (* Michael De Vlieger, Apr 19 2015 *)
Table[PowerMod[n,2,NextPrime[n]],{n,90}] (* Harvey P. Dale, May 24 2015 *)

PARI

a(n)=n^2%nextprime(n+1)