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A214583
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Numbers m such that for all k with gcd(m, k) = 1 and m > k^2, m - k^2 is prime.
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6
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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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refs;
listen;
history;
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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, ", ") );
); }
(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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