A192864 -id:A192864 - cp's OEIS frontend

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

3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 73, 79, 83, 101, 103, 109, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 181, 191, 193, 211, 223, 227, 229, 239, 257, 263, 271, 277, 281, 283, 307, 311, 313, 317, 331, 347, 353, 367, 373, 379

Offset: 1

Charles R Greathouse IV, Jul 11 2011

nonn

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.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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)

Subsequence of A192861.

Cf. A192863, A192864, A005596.

is(n)=issquarefree((n+1)>>valuation(n+1,2)) && isprime(n) && n>2 \\ Charles R Greathouse IV, Mar 26 2014

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

A192863 Lower flat numbers: odd numbers k such that k-1 is a squarefree number times a power of two.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 47, 49, 53, 57, 59, 61, 63, 65, 67, 69, 71, 75, 77, 79, 81, 83, 85, 87, 89, 93, 95, 97, 103, 105, 107, 111, 113, 115, 117, 119, 121, 123, 125, 129, 131, 133, 135, 137, 139, 141, 143, 147

Offset: 1

Charles R Greathouse IV, Jul 11 2011

nonn

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.

Cf. A192861, A192862, A192864.

Cf. A185199 (asymptotic density of this sequence).

Select[Range[3, 150, 2], SquareFreeQ[(# - 1)/2^IntegerExponent[# - 1, 2]] &] (* Amiram Eldar, Aug 30 2020 *)

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

a(n) ~ Pi^2/4 * n.

Data corrected by Amiram Eldar, Aug 30 2020

cp's OEIS Frontend

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

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