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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 (list; graph; refs; listen; history; text; internal format)
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

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 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 *)

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. 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

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Last modified November 20 10:07 EST 2017. Contains 294963 sequences.