

A214583


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


5



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
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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 (96825*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

Table of n, a(n) for n=1..41.
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 201*1 = 19 (prime), and 203*3 = 11 (prime). Not considered are 202*2 and 204*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]] (* JeanFrançois Alcover, Oct 11 2013, after Joerg Arndt's Pari program *)


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(nk^2), q=0; break() );
);
k += 1;
);
if ( q, print1(n, ", ") );
); }
/* Joerg Arndt, Jul 21 2012 */
(Haskell)
a214583 n = a214583_list !! (n1)
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. A057828, A000290, A010051.
Cf. A224075; subsequence of A008864.
Sequence in context: A243653 A203444 A008864 * A232721 A227956 A225531
Adjacent sequences: A214580 A214581 A214582 * A214584 A214585 A214586


KEYWORD

nonn,nice


AUTHOR

Hans Ruegg, Jul 21 2012


STATUS

approved



