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Thin numbers: odd numbers k > 1 such that k+1 is a prime times a power of two.
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%I #14 Aug 30 2020 03:25:14

%S 3,5,7,9,11,13,15,19,21,23,25,27,31,33,37,39,43,45,47,51,55,57,61,63,

%T 67,73,75,79,81,85,87,91,93,95,103,105,111,115,117,121,123,127,133,

%U 135,141,145,147,151,157,159,163,165,171,175,177,183,187,191,193

%N Thin numbers: odd numbers k > 1 such that k+1 is a prime times a power of two.

%H Amiram Eldar, <a href="/A192868/b192868.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) ~ n log n.

%t Select[Range[3, 200, 2], (r = (# + 1)/2^IntegerExponent[# + 1, 2]) == 1 || PrimeQ[r] &] (* _Amiram Eldar_, Aug 30 2020 *)

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

%Y Cf. A192869.

%K nonn

%O 1,1

%A _Charles R Greathouse IV_, Jul 11 2011