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

Author

Charles R Greathouse IV, Jul 11 2011

Keywords

• nonn

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)

Crossrefs

Subsequence of A192868.

Programs

Mathematica

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

PARI

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

Formula

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