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A214583
Numbers m such that for all k with gcd(m, k) = 1 and m > k^2, m - k^2 is prime.
6
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
1,1
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
LINKS
Leo Moser, An Introduction to the Theory of Numbers, The Trillia Group 2004, page 84.
EXAMPLE
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.
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 *)
PROG
(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 */
(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
CROSSREFS
Cf. A065428.
Cf. A224075; subsequence of A008864.
Sequence in context: A203444 A008864 A299763 * A232721 A227956 A320592
KEYWORD
nonn,nice
AUTHOR
Hans Ruegg, Jul 21 2012
STATUS
approved