A085099 Least natural number k such that k^2 + n is prime.

1, 1, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 4, 1, 6, 7, 2, 9, 2, 1, 12, 1, 4, 3, 2, 3, 6, 1, 2, 3, 2, 1, 24, 1, 2, 3, 4, 1, 6, 5, 2, 3, 4, 1, 6, 5, 2, 9, 2, 1, 18, 1, 6, 3, 2, 3, 6, 1, 2, 9, 2, 1, 6, 1, 4, 3, 2, 5, 6, 1, 2, 3, 4, 1, 12, 5, 2

Offset: 1

Author

Jason Earls, Aug 10 2003

Keywords

• nonn
• easy

Comments

First values of k and n such that k > 100 are: k=114, n=6041; for k > 200: k=210, n=26171; for k > 300: k=357, n=218084; for k > 400: k=402, n=576239.

Additionally, for k > 500: k=585, n=3569114; for k > 600: k=630, n=3802301; for k > 700: k=744, n=24307841; for k > 800: k=855, n=25051934; for k > 900: k=1008, n=54168539. Other cases k > 900: k=945, n=74380946, k=915, n=89992964, k=939, n=118991066. - Zak Seidov, May 23 2007

It is easily proved that for n > 2, a(n) >= A089128(n+1). The first inequality is a(21) = 4. - Franklin T. Adams-Watters, May 16 2018

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A059843, A200926, A058055, A053000, A089128.

Programs

Maple

a:= proc(n) local d, t; d, t:= 1, n+1; while not
      isprime(t) do d:= d+2; t:= t+d od; (d+1)/2
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 04 2019

Mathematica

Table[i = 1; While[! PrimeQ[i^2 + n], i++]; i, {n, 85}] (* Jayanta Basu, Apr 24 2013 *)

PARI

a(n)=my(k); while(!isprime(k++^2+n),); k \\ Charles R Greathouse IV, Jul 17 2016

Formula

a(n) = sqrt(A059843(n) - n). - Zak Seidov, Nov 24 2011