A192864 Lower flat primes: odd primes p such that p-1 is a squarefree number times a power of two.

3, 5, 7, 11, 13, 17, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 103, 107, 113, 131, 137, 139, 149, 157, 167, 173, 179, 191, 193, 211, 223, 227, 229, 233, 239, 241, 257, 263, 269, 277, 281, 283, 293, 311, 313, 317, 331, 337, 347, 349, 353, 359

Offset: 1

Author

Charles R Greathouse IV, Jul 11 2011

Keywords

• nonn

Comments

Broughan & Qizhi show that this sequence has relative density 2*A in the primes, where A = A005596 is Artin's constant. Consequently, there exists a flat number between x and (1+e)x for every e > 0 and large enough x.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Kevin A. Broughan and Zhou Qizhi, Flat primes and thin primes, Bulletin of the Australian Mathematical Society, Vol. 82, No. 2 (2010), pp. 282-292, alternative link.

Crossrefs

Subsequence of A192863.

Cf. A192861, A192862, A005596.

Programs

Mathematica

Select[Range[3, 360, 2], PrimeQ[#] && SquareFreeQ[(# - 1)/2^IntegerExponent[# - 1, 2]] &] (* Amiram Eldar, Aug 30 2020 *)

PARI

is(n)=n%2&&isprime(n)&&issquarefree((n-1)>>valuation(n-1,2)) \\ corrected by Amiram Eldar, Aug 30 2020

Formula

a(n) ~ k * n * log(n) with k = 1/(2*A) = 1.3370563...

Extensions

Data corrected by Amiram Eldar, Aug 30 2020