A214583 Numbers m such that for all k with gcd(m, k) = 1 and m > k^2, m - k^2 is prime.

3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 48, 54, 60, 62, 68, 72, 80, 84, 90, 98, 108, 110, 132, 138, 140, 150, 180, 182, 198, 252, 318, 360, 398, 468, 570, 572, 930, 1722

Offset: 1

Author

Hans Ruegg, Jul 21 2012

Keywords

• nonn
• nice

Comments

No further terms < 10^10.

This sequence is based on a remark in a paper distributed over the Internet (see the Leo Moser link) under the heading "Unsolved Problems and Conjectures" (page 84):

"Is 968 the largest number n such that for all k with (n, k) = 1 and n > k^2, n - k^2 is prime? (Erdős)"

The statement by Moser contains an error: 968 does NOT have this property (968-25*25 = 343 = 7*7*7), and the largest such number (1722) is larger than 968.

A224076(n) <= A064272(a(n)+1). - Reinhard Zumkeller, Mar 31 2013

Examples

			For example, the number 20 is part of this sequence because 20-1*1 = 19 (prime), and 20-3*3 = 11 (prime). Not considered are 20-2*2 and 20-4*4, because 2 and 4 are not relative primes to 20.
		

Links

• Leo Moser, An Introduction to the Theory of Numbers, The Trillia Group 2004, page 84.

Crossrefs

Cf. A065428.

Cf. A057828, A000290, A010051.

Cf. A224075; subsequence of A008864.

Programs

Haskell

a214583 n = a214583_list !! (n-1)
a214583_list = filter (p 3 1) [2..] where
   p i k2 x = x <= k2 || (gcd k2 x > 1 || a010051' (x - k2) == 1) &&
                         p (i + 2) (k2 + i) x
-- Reinhard Zumkeller, Mar 31 2013, Jul 22 2012

Mathematica

Reap[For[p = 2, p < 2000, p = NextPrime[p], n = p+1; q = True; k = 1; While[k*k < n, If[GCD[k, n] == 1, If[! PrimeQ[n - k^2], q = False; Break[]]]; k += 1]; If[q, Sow[n]]]] [[2, 1]] (* Jean-François Alcover, Oct 11 2013, after Joerg Arndt's Pari program *)
gQ[n_]:=AllTrue[n-#^2&/@Select[Range[Floor[Sqrt[n]]],CoprimeQ[ #, n]&], PrimeQ]; Select[Range[2000],gQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 02 2018 *)

PARI

N=10^10;
default(primelimit,N);
{ forprime (p=2, N,
    n = p + 1;
    q = 1;
    k = 1;
    while ( k*k < n,
        if ( gcd(k,n)==1,
            if ( ! isprime(n-k^2),  q=0; break() );
        );
        k += 1;
    );
    if ( q, print1(n,", ") );
); }
/* Joerg Arndt, Jul 21 2012 */