A126715 -id:A126715 - cp's OEIS frontend

A247202 Smallest odd k > 1 such that k*2^n - 1 is a prime number.

3, 3, 3, 3, 7, 3, 3, 5, 7, 5, 3, 5, 9, 5, 9, 17, 7, 3, 51, 17, 7, 33, 13, 39, 57, 11, 21, 27, 7, 213, 15, 5, 31, 3, 25, 17, 21, 3, 25, 107, 15, 33, 3, 35, 7, 23, 31, 5, 19, 11, 21, 65, 147, 5, 3, 33, 51, 77, 45, 17, 69, 53, 9, 3, 67, 63, 43, 63, 51, 27, 73, 5

Offset: 1

Pierre CAMI, Nov 25 2014

nonn

Limit_{N->oo} (Sum_{n=1..N} a(n))/(Sum_{n=1..N} n) = log(2). [[Is there a proof or is this a conjecture? - Peter Luschny, Feb 06 2015]]

Records: 3, 7, 9, 17, 51, 57, 213, 255, 267, 321, 615, 651, 867, 901, 909, 1001, 1255, 1729, 1905, 2163, 3003, 3007, 3515, 3797, 3825, 4261, 4335, 5425, 5717, 6233, 6525, 6763, 11413, 11919, 12935, 20475, 20869, 25845, 30695, 31039, 31309, 42991, 55999, ... . - Robert G. Wilson v, Feb 08 2015

Robert G. Wilson v, Table of n, a(n) for n = 1..10031 (first 5000 terms from Pierre CAMI)

Cf. A126715, A247479.

f:= proc(n)
local k,p;
  p:= 2^n;
for k from 3 by 2 do if isprime(k*p-1) then return k fi od;
end proc:
seq(f(n), n=1 .. 100); # Robert Israel, Feb 05 2015

f[n_] := Block[{k = 3, p = 2^n}, While[ !PrimeQ[k*p - 1], k += 2]; k]; Array[f, 70]

a(n) = {k=3; while (!isprime(k*2^n-1), k+=2); k;} \\ Michel Marcus, Nov 25 2014

a(A002235(n)) = 3.

A128979 Least exponent k such that p_n*(2^k) - 1 is prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 4, 4, 1, 1, 2, 7, 4, 2, 12, 3, 5, 2, 7, 1, 2, 4, 1, 10, 3, 10, 9, 8, 25, 2, 2, 1, 4, 5, 1, 3, 4, 2, 8, 3, 226, 3, 2, 1, 1, 3, 2, 1, 4, 4, 11, 6, 4, 2, 8, 1, 5, 2, 11, 2, 1, 26, 3, 6, 1, 1, 18, 3, 4, 4, 1, 7, 1, 2, 20, 5, 10, 3, 4, 7, 2, 3, 1, 6, 112, 9, 10, 7, 2, 12, 5, 46, 1, 2, 8

Offset: 1

Pierre CAMI and Robert G. Wilson v, Feb 16 2007

nonn

Supposedly the difference from A101050 is that the k here are required to be strictly positive (nonzero positive). - R. J. Mathar, Dec 13 2008

Pierre CAMI and Robert G. Wilson v, Table of n, a(n) for n = 1..119.

Cf. A000040, A101050, A126715.

f[n_] := Block[{k = 1, p = Prime@n}, While[ !PrimeQ[p*2^k - 1], k++ ]; k]; Array[f, 97]

A245441 a(1)=3, then a(n) = smallest odd k > Ceiling(a(n-1)/2) such that k*2^n-1 is prime.

Original entry on oeis.org

3, 3, 3, 3, 7, 17, 13, 27, 25, 15, 25, 23, 21, 15, 9, 17, 15, 21, 51, 35, 19, 33, 25, 39, 57, 57, 81, 45, 45, 213, 111, 57, 31, 131, 99, 83, 45, 27, 25, 107, 55, 33, 33, 35, 67, 141, 91, 89, 69, 41, 129, 89, 147, 101, 195, 129, 79, 77, 45, 77, 69, 53, 61

Offset: 1

Pierre CAMI, Jul 22 2014

nonn

A126715(n) = smallest odd k such that k*2^n-1 is prime, the primes are not always in increasing order.
Here the primes k*2^n-1 are always in increasing order.
The ratio sum_{1..N}a(n)/sum_{1..N}n is near 2*log(2) as N increases.
The ratio a(n)/n is always < 8 for n from 1 to 6000.

			3*2^1-1 = 5 is prime, a(1)=3 by definition.
3*2^2-1 = 11 is prime, 3 > 3/2 so a(2) = 3.
3*2^3-1 = 23 is prime, so a(3) = 3.
3*2^4-1 = 47 is prime, so a(4) = 3.
3*2^5-1 = 95 is composite.
5*2^5-1 = 159 is composite.
7*2^5-1 = 223 is prime so a(5) = 7.

Pierre CAMI, Table of n, a(n) for n = 1..6000

Cf. A126715.

a=[3]; for(n=2, 100, k=floor(a[n-1]/2)+2; if(k%2==0, k++); t=2^n; while(!isprime(k*t-1), k+=2); a=concat(a, k)); a \\ Colin Barker, Jul 22 2014

A179289 Smallest index k such that prime(k)*2^n-1 is prime, or zero if there is no prime.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 2, 3, 4, 3, 2, 1, 11, 3, 22, 1, 4, 1, 18, 7, 4, 23, 6, 23, 18, 5, 44, 23, 4, 1, 14, 3, 11, 2, 11, 7, 11, 2, 18, 28, 8, 16, 2, 102, 4, 9, 11, 3, 8, 5, 174, 24, 63, 3, 2, 103, 22, 23, 130, 1, 22, 16, 18, 2, 19, 55, 14, 41, 34, 15

Offset: 1

Pierre CAMI, Jul 09 2010

nonn

Define partial sum S(N)=Sum(n=1...N) of n , T(N)=Sum(n=1...N) of k(n) The ratio T(N)/S(N) --> approx 0.5236 as N --> infinity.

It is conjectured that a(42228) is the first 0 term. This corresponds to the first Riesel number, 509203, which happens to be prime. See A101036. - T. D. Noe, Mar 23 2011

Cf. A126715.

sik[n_]:=Module[{c=2^n,k=1},While[!PrimeQ[Prime[k]*c-1],k++];k]; Array[ sik,70] (* The program will NOT identify cases where no prime satisfies the definition.  See the second comment. *) (* Harvey P. Dale, Jan 10 2016 *)

SCRIPT / DIM nn,0 / DIM kk / DIMS st / LABEL loopn / SET nn,nn+1 / IF nn>10000 THEN END / SET kk,0 / LABEL loopk / SET kk,kk+1 / SET st,%d,%d,%d\,;nn;kk;p(kk) / PRP p(kk)*2^nn-1 / IF ISPRIME THEN GOTO loopn / GOTO loopk / / This file is the in.txt file / The command is PFGW -f in.txt / The results are in the file pfgw-prime.log for small n / and in the pfgw.log file for greatest n / Program PFGW from Primeform Group /

a(n) = {my(k=1); while (!isprime(prime(k)*2^n-1), k++); k;} \\ Michel Marcus, Sep 16 2019

a(n) = 1 for n = A000043(k) - 1, the Mersenne exponents minus 1. - T. D. Noe, Mar 23 2011

A187467 Least k > 1 such that prime(k)*2^n - 1 is prime, or zero if never prime.

Original entry on oeis.org

2, 2, 2, 2, 4, 2, 2, 3, 4, 3, 2, 3, 11, 3, 22, 7, 4, 2, 18, 7, 4, 23, 6, 23, 18, 5, 44, 23, 4, 98, 14, 3, 11, 2, 11, 7, 11, 2, 18, 28, 8, 16, 2, 102, 4, 9, 11, 3, 8, 5, 174, 24, 63, 3, 2, 103, 22, 23, 130, 7, 22, 16, 18, 2

Offset: 1

Pierre CAMI, Mar 10 2011

nonn

As N increases, it appears that (Sum_{i=1..N} a(i)) / (Sum_{i=1..N} i) tends to 1/2, i.e., the partial sums grow roughly proportional to the triangular numbers.

It is conjectured that a(42228) is the first 0 term. This corresponds to the first Riesel number, 509203, which happens to be prime. See A101036. - T. D. Noe, Mar 23 2011

Pierre CAMI, Table of n, a(n) for n = 1..4100

Cf. A101036, A126715, A187468.

A187467 := proc(n) local k; for k from 2 do if isprime( ithprime(k)*2^n-1) then return k; end if; end do: end proc: # R. J. Mathar, Mar 19 2011

a(n) = primepi(A126715(n)). - T. D. Noe, Mar 10 2011

a(n) >= A179289(n). - R. J. Mathar, Mar 19 2011

cp's OEIS Frontend

A247202 Smallest odd k > 1 such that k*2^n - 1 is a prime number.

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A128979 Least exponent k such that p_n*(2^k) - 1 is prime.

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A245441 a(1)=3, then a(n) = smallest odd k > Ceiling(a(n-1)/2) such that k*2^n-1 is prime.

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A179289 Smallest index k such that prime(k)*2^n-1 is prime, or zero if there is no prime.

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A187467 Least k > 1 such that prime(k)*2^n - 1 is prime, or zero if never prime.

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