A192862 -id:A192862 - cp's OEIS frontend

A192861 Flat numbers: odd n such that n+1 is a squarefree number times a power of two.

1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 51, 55, 57, 59, 61, 63, 65, 67, 69, 73, 75, 77, 79, 81, 83, 85, 87, 91, 93, 95, 101, 103, 105, 109, 111, 113, 115, 117, 119, 121, 123, 127, 129, 131, 133, 135, 137, 139, 141

Offset: 1

Charles R Greathouse IV, Jul 11 2011

nonn

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.

Cf. A192862.

is(n)=n%2&&issquarefree((n+1)>>valuation(n+1,2))

list(lim)=my(v=List()); for(k=1,logint(1+lim\=1,2), forsquarefree(n=1,(lim+1)>>k, listput(v, n[1]<Charles R Greathouse IV, Mar 01 2018

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

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

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

Original entry on oeis.org

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

Charles R Greathouse IV, Jul 11 2011

nonn

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.

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.

Subsequence of A192863.

Cf. A192861, A192862, A005596.

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

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

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

Data corrected by Amiram Eldar, Aug 30 2020

cp's OEIS Frontend

A192861 Flat numbers: odd n such that n+1 is a squarefree number times a power of two.

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

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

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Crossrefs

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Mathematica

PARI

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Extensions