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Lower flat primes: odd primes p such that p-1 is a squarefree number times a power of two.
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%I #12 Sep 01 2020 08:55:35

%S 3,5,7,11,13,17,23,29,31,41,43,47,53,59,61,67,71,79,83,89,97,103,107,

%T 113,131,137,139,149,157,167,173,179,191,193,211,223,227,229,233,239,

%U 241,257,263,269,277,281,283,293,311,313,317,331,337,347,349,353,359

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

%C 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.

%H Amiram Eldar, <a href="/A192864/b192864.txt">Table of n, a(n) for n = 1..10000</a>

%H Kevin A. Broughan and Zhou Qizhi, <a href="https://doi.org/10.1017/S0004972710000067">Flat primes and thin primes</a>, Bulletin of the Australian Mathematical Society, Vol. 82, No. 2 (2010), pp. 282-292, <a href="http://www.math.waikato.ac.nz/~kab/papers/flatandthin4.pdf">alternative link</a>.

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

%t Select[Range[3, 360, 2], PrimeQ[#] && SquareFreeQ[(# - 1)/2^IntegerExponent[# - 1, 2]] &] (* _Amiram Eldar_, Aug 30 2020 *)

%o (PARI) is(n)=n%2&&isprime(n)&&issquarefree((n-1)>>valuation(n-1,2)) \\ corrected by _Amiram Eldar_, Aug 30 2020

%Y Subsequence of A192863.

%Y Cf. A192861, A192862, A005596.

%K nonn

%O 1,1

%A _Charles R Greathouse IV_, Jul 11 2011

%E Data corrected by _Amiram Eldar_, Aug 30 2020