A192869 Thin primes: odd primes p such that p+1 is a prime (or 1) times a power of two.

3, 5, 7, 11, 13, 19, 23, 31, 37, 43, 47, 61, 67, 73, 79, 103, 127, 151, 157, 163, 191, 193, 211, 223, 271, 277, 283, 313, 331, 367, 383, 397, 421, 457, 463, 487, 523, 541, 547, 607, 613, 631, 661, 673, 691, 733, 751, 757, 787, 823, 877, 907, 991, 997, 1051

Offset

1,1

Comments

Broughan & Qizhi conjecture that a(n) << n (log n)^2, matching the lower bound they proved.

Sequence A206581 excludes the Mersenne primes (A000043), which are included here under the "or 1" case. - T. D. Noe, Mar 07 2012

References

D. R. Heath-Brown, "Artin's conjecture for primitive roots", Quarterly Journal of Mathematics 37:1 (1986) pp. 27-38.

N. M. Timofeev, "The Hardy-Ramanujan and Halasz inequalities for shifted primes", Mathematical Notes 57:5 (1995), pp. 522-535.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Kevin Broughan and Zhou Qizhi, Flat primes and thin primes, Bulletin of the Australian Mathematical Society 82:2 (2010), pp. 282-292.

Qizhi Zhou, Multiply perfect numbers of low abundancy, PhD thesis (2010)

Formula

a(n) >> n (log n)^2.

Mathematica

onePrimeQ[n_] := n == 1 || PrimeQ[n]; Select[Prime[Range[2, 1000]], onePrimeQ[(# + 1)/2^IntegerExponent[# + 1, 2]] &] (* T. D. Noe, Mar 06 2012 *)

Prog

(PARI) is(n)=n%2&&isprime(n)&&(isprime((n+1)>>valuation(n+1, 2)) || n+1==1<<valuation(n+1, 2))

Crossrefs

Subsequence of A192868.

Sequence in context: A216371 A095747 A132779 * A147513 A075323 A020575

Adjacent sequences: A192866 A192867 A192868 * A192870 A192871 A192872

Keyword

nonn

Author

Charles R Greathouse IV, Jul 11 2011

Status

approved