A257231 - OEIS

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

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OFFSET

1,3

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).

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

EXAMPLE

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

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

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 *)

PROG

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