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A156620 Primes p such that p^2 - 2 is a 5-almost prime. 1
1201, 2999, 4001, 4273, 5009, 7151, 8467, 9769, 10427, 10937, 11701, 11897, 12011, 12113, 12323, 13339, 13681, 14087, 14563, 15187, 15277, 15809, 16139, 16699, 17209, 17383, 17483, 17623, 18757, 19051, 19267, 19697, 20107, 20129, 20297 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Corresponding 5-almost primes are A156621.

This sequence is infinite: Ribenboim states that Rieger proved in 1969 that "there exist infinitely many primes p such that p^2 - 2 [is an element of] P_5", this being a particular case of a general theorem proved (also in 1969) by Richert: (again quoting Ribenboim) "Let f(X) be a polynomial with integral coefficients, positive leading coefficient, degree d >= 1 (and different from X). Assume that for every prime p, the number [rho](p) of solutions of f(X) = 0 (mod p) is less than p; moreover if p <= d+1 and p does not divide f(0) assume also that [rho](p) < p-1. Then, there exist infinitely many primes p such that f(p) is a (2d+1)-almost prime."

REFERENCES

H. Halberstam and H. E. Richert, Sieve Methods, Academic Press, NY, 1974.

P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 184.

G. J. Rieger, On polynomials and almost-primes, Bull. Amer. Math. Soc., 75 (1969), 100-103.

LINKS

Table of n, a(n) for n=1..35.

MATHEMATICA

Select[Prime[Range[5000]], PrimeOmega[#^2-2]==5&] (* Harvey P. Dale, Jul 11 2014 *)

PROG

(PARI) forprime(p=2, prime(2500), if(bigomega(p^2-2)==5, print1(p, ", ")))

CROSSREFS

Cf. A156621, A014614.

Sequence in context: A282015 A217656 A020390 * A214116 A221451 A133142

Adjacent sequences: A156617 A156618 A156619 * A156621 A156622 A156623

KEYWORD

nonn

AUTHOR

Rick L. Shepherd, Feb 11 2009

STATUS

approved

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Last modified December 1 18:31 EST 2022. Contains 358475 sequences. (Running on oeis4.)