A192861 -id:A192861 - cp's OEIS frontend

A204458 Odd numbers not divisible by 17.

1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141

Offset: 1

Wolfdieter Lang, Feb 07 2012

For the general case of odd numbers not divisible by a prime see a comment on A204454. There the o.g.f.s and the formulas are given.
The numerator polynomial of the o.g.f. given below has coefficients 1,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,1. See the row no. 7 of the array A204456. The first nine numbers are the first differences of the sequence if one starts with a(0):=0. The remaining ones are obtained by mirroring around the central number 4.
Compare with A192861: certain numbers from here are missing there, like 35, 49, 53, 71, 89, 97, 99, .. and others are missing here like 51, 85, 119, ...
Numbers coprime to 34. The asymptotic density of this sequence is 8/17. - Amiram Eldar, Oct 20 2020

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A204454 (also for more crossrefs), A204457.

Select[Range[141], CoprimeQ[#, 34] &] (* Amiram Eldar, Oct 20 2020 *)

O.g.f.: x*(1 + x^16 + 2*x*(1+x^8)*(Sum_{k=0..6} x^k) + 4*x^8)/((1-x^16)*(1-x)). The denominator can be factored.
a(n) = 2*n-1 + 2*floor((n+7)/16) = 2*n+1 + 2*floor((n-9)/16), n>=1. Note that for n=0 this is -1, but for the o.g.f. with start x^0 one uses a(0)=0.
a(n) = a(n-1) + a(n-16) - a(n-17). - Wesley Ivan Hurt, Oct 20 2020

A192862 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, 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

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

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A204458 Odd numbers not divisible by 17.

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A192862 Flat primes: odd primes p such that p+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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Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions