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 A120627 Least positive k such that both prime(n)+k and prime(n)+2k are prime, or 0 if no such k exists. 6
 0, 2, 6, 6, 6, 24, 6, 12, 18, 12, 6, 30, 6, 18, 6, 18, 12, 6, 6, 18, 54, 24, 24, 12, 6, 6, 24, 30, 42, 18, 12, 18, 30, 12, 24, 6, 36, 18, 6, 54, 84, 30, 36, 18, 30, 12, 30, 54, 6, 42, 18, 12, 36, 6, 6, 48, 12, 6, 30, 36, 24, 54, 30, 36, 18, 36, 18, 30, 6, 24, 48, 30, 6, 24, 30, 18, 30 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Note that 6 divides a(n) for n>2. - T. D. Noe, Aug 29 2006 Van der Corput's theorem: There are infinitely many positive integers n, k such that n, n+nk, n+2nk are all prime. - Jonathan Vos Post, Apr 17 2007 REFERENCES A. G. van der Corput, Uber Summen von Primzahlen und Primzahlquadraten," Math. Ann., 116 (1939) 1-50. LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 Terence Tao, Simons Lecture I: Structure and randomness in Fourier analysis and number theory, April 2007. EXAMPLE a(3)=6 because prime(3)=5 and 5+6 and 5+12 are primes. MATHEMATICA f[n_] := Block[{p = Prime[n], k = 1}, If[n == 1, 0, While[ ! PrimeQ[p + 2k] || ! PrimeQ[p + 4k], k++ ]; 2k] ]; Table[f[n], {n, 80}] (*Chandler*) Join[{0}, Table[p=Prime[n]; k=2; While[ !PrimeQ[p+k] || !PrimeQ[p+2k], k=k+2]; k, {n, 2, 100}]] - T. D. Noe, Aug 29 2006 PROG (PARI) a(n)=if(n<2, 0, my(p=prime(n), k); while(!isprime(p+k++)||!isprime(p+2*k), ); k) \\ Charles R Greathouse IV, Apr 24 2015 CROSSREFS Cf. A000040. Sequence in context: A117217 A260930 A161331 * A089879 A087651 A078579 Adjacent sequences:  A120624 A120625 A120626 * A120628 A120629 A120630 KEYWORD easy,nonn AUTHOR Giovanni Teofilatto, Aug 25 2006 EXTENSIONS Edited and extended by Ray Chandler and T. D. Noe, Aug 28 2006 STATUS approved

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