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A192862 Flat primes: odd primes p such that p+1 is a squarefree number times a power of two. 4


%S 3,5,7,11,13,19,23,29,31,37,41,43,47,59,61,67,73,79,83,101,103,109,

%T 113,127,131,137,139,151,157,163,167,173,181,191,193,211,223,227,229,

%U 239,257,263,271,277,281,283,307,311,313,317,331,347,353,367,373,379

%N 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 2A 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 Charles R Greathouse IV, <a href="/A192862/b192862.txt">Table of n, a(n) for n = 1..10000</a>

%H Kevin Broughan and Zhou Qizhi, <a href="http://www.math.waikato.ac.nz/~kab/papers/flatandthin4.pdf">Flat primes and thin primes</a>, Bulletin of the Australian Mathematical Society 82:2 (2010), pp. 282-292.

%H Qizhi Zhou, <a href="http://researchcommons.waikato.ac.nz/handle/10289/4138">Multiply perfect numbers of low abundancy</a>, PhD thesis (2010)

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

%o (PARI) is(n)=issquarefree((n+1)>>valuation(n+1,2)) && isprime(n) && n>2 \\ _Charles R Greathouse IV_, Mar 26 2014

%Y Subsequence of A192861.

%Y Cf. A192863, A192864, A005596.

%K nonn

%O 1,1

%A _Charles R Greathouse IV_, Jul 11 2011

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Last modified September 22 19:23 EDT 2021. Contains 347608 sequences. (Running on oeis4.)