A050412 -id:A050412 - cp's OEIS frontend

A050544 Numbers k such that 37*2^k-1 is prime.

1, 2553, 4893, 7897, 14353, 32125, 51477, 78973, 341365

Offset: 1

N. J. A. Sloane, Dec 29 1999

Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime
Kosmaj, Riesel list k<300.

is(n)=ispseudoprime(37*2^n-1) \\ Charles R Greathouse IV, Jun 12 2017

More terms from David Broadhurst, Dec 29 1999

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008

A078681 Start with n; iterate the process x -> ceiling(3x/2) until reach a prime. Sequence gives number of steps to reach a prime or 0 if no prime is ever reached.

Original entry on oeis.org

1, 1, 1, 53, 5, 52, 1, 4, 51, 2, 1, 3, 39, 50, 1, 162, 3, 2, 1, 38, 49, 4, 2, 161, 50, 2, 1, 52, 4, 37, 1, 48, 3, 13, 1, 160, 9, 49, 1, 8, 42, 51, 18, 3, 36, 8, 1, 47, 3, 2, 12, 35, 27, 159, 1, 8, 48, 2, 1, 7, 4, 41, 50, 16, 17, 2, 1, 35, 7, 13, 1, 46, 11, 2, 1, 3, 11, 34, 2, 26, 158, 7, 6, 7

Offset: 1

Benoit Cloitre, Dec 17 2002

nonn

Cf. A050412.

a(n)=if(n<0,0,s=n; c=1; while(isprime(ceil(3*s/2))==0,s=ceil(3*s/2); c++); c)

A138508 Semiprime analog of Riesel problem: start with n; repeatedly double and add 1 until reach a semiprime. Sequence gives number of steps to reach a semiprime or 0 if no semiprime is ever reached.

Original entry on oeis.org

3, 5, 2, 1, 4, 3, 1, 2, 2, 1, 3, 1, 2, 3, 5, 1, 1, 4, 1, 4, 2, 2, 2, 1, 1, 3, 1, 1, 2, 2, 4, 1, 4, 1, 2, 3, 3, 1, 2, 3, 3, 1, 1, 7, 1, 1, 1, 3, 4, 2, 3, 10, 2, 2, 1, 6, 1, 2, 1, 1, 1, 4, 3, 1, 3, 1, 3, 3, 3, 1, 1, 1, 2, 2, 2, 5, 1, 4, 1, 1, 2, 6, 2, 1, 3, 3, 3, 1, 6, 5, 1, 1, 1, 5, 3, 5, 2, 2, 3, 1, 1, 1, 2, 1, 9, 1, 1, 1, 1, 1, 2

Offset: 1

Jonathan Vos Post, May 10 2008

This is the analog of A050412 with "prime" replaced by "semiprime". [Edited by Felix Fröhlich, Apr 21 2021]
a(n) is the smallest m>=0 such that (n+1)*2^m-1 is semiprime, or 0 if no such semiprime exists. - R. J. Mathar, May 12 2008
There is no "semiprime Riesel number" (i.e., n such that a(n) = 0) among all n up to 2*10^6. - Felix Fröhlich, Apr 21 2021

Cf. A001358, A050412.

isA001358 := proc(n) RETURN( numtheory[bigomega](n) = 2) ; end:
A138508 := proc(n) local a,niter ; niter := n ; a := 0 ; while not isA001358(niter) do a := a+1 ; niter := 2*niter+1 ; od: a ; end:
seq(A138508(n),n=1..200) ; # R. J. Mathar, May 12 2008

a(n) = my(x=n, i=0); while(1, x=2*x+1; i++; if(bigomega(x)==2, return(i))); \\ Felix Fröhlich, Apr 21 2021

More terms from R. J. Mathar, May 12 2008

All terms corrected by Felix Fröhlich, Apr 21 2021

A225721 Starting with x = n, the number of iterations of x := 2x - 1 until x is prime, or -1 if no prime exists.

Original entry on oeis.org

-1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 6, 1, 1, 0, 1, 2, 2, 1, 2, 0, 1, 0, 8, 3, 1, 2, 1, 0, 2, 5, 1, 0, 1, 0, 2, 1, 2, 0, 583, 1, 2, 1, 1, 0, 1, 1, 4, 1, 2, 0, 5, 0, 4, 7, 1, 2, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 1, 4, 3, 0, 2, 3, 1, 0, 1, 2, 4

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 13 2013

sign

This appears to be a shifted variant of A040076. - R. J. Mathar, May 28 2013

If n is prime, then a(n) = 0. If the sequence never reaches a prime number (for n = 1) or the prime number has more than 1000 digits, -1 is used instead. There are 22 such numbers for n < 10000.

			For a(20), the trajectory is 20->39->77->153->305->609->1217, a prime number. That required 6 steps, so a(20)=6.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A050921 (primes obtained).
Cf. A040081, A038699, A050412, A052333, A046069 (related to the Riesel problem).
Cf. A000668, A000043, A065341 (Mersenne primes), A000079 (powers of 2).
Cf. A007770 (happy numbers), A031177 (unhappy numbers).
Cf. A037274 (home primes), A037271 (steps), A037272, A037272.

y=as.bigz(rep(0,500)); ys=rep(0,500);
for(i in 1:500) { n=as.bigz(i); k=0;
    while(isprime(n)==0 & ndig(n)<1000 & k<5000) { k=k+1; n=2*n-1 }
    if(ndig(n)>=1000 | k>=5000) { ys[i]=-1; y[i]=-1;
    } else {ys[i]=k; y[i]=n; }
}

A257495 The number of iterations (x -> 2x+1) until a prime is found, starting with prime(n); or 0 if a prime is never found.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 2, 2, 1, 1, 2, 2, 1, 24, 2, 1, 2, 4, 2, 4, 2552, 4, 1, 1, 4, 8, 4, 2, 2, 1, 6, 1, 3, 4, 2, 2, 2, 8, 4, 1, 1, 2, 1, 8, 3, 6, 4, 4, 2, 2, 1, 1, 2, 1, 2, 3, 8, 2, 4, 1, 12, 1, 2, 21, 4, 3, 2, 4, 6, 2, 11, 1, 2, 16, 4, 4, 2, 4, 2, 8, 1, 12, 1, 8

Offset: 1

Bill McEachen, Apr 26 2015

nonn

The number of iterations is defined as in A050412 (probably always positive).
Sophie Germain primes correspond to values a(n)=1 (A156660).
The plot without largest outliers allows detail on lower bound trending. Such outliers begin beyond the 121st entry. The number of terminal primes (those from the terminating iteration) being Sophie Germain through the first 10000 seeds is approximately 910. The number of Sophie Germain primes expected below 10000 is approximately 156 (computationally the comparison is more complicated, obviously).
For prime(21) = 73, a(21) = 2552 corresponds to the prime 12525084203....315016703 with 771 digits. See A171390. - Vincenzo Librandi, Apr 27 2015
a(n) is the smallest k > 0 such that (prime(n) + 1)*2^k - 1 is prime. - Thomas Ordowski, Jun 05 2019

			Starting from prime(6)=13, sequential values for evaluation are 2*13+1=27, 2*27+1=55, 2*55+1=111, 2*111+1=223. The first prime is encountered at the 4th iteration, thus a(6)=4.

Bill McEachen, Table of n, a(n) for n = 1..7075, using ispseudoprime() in the Pari code.

Cf. A050412 (Riesel problem).

A257495 := proc(n)
    A050412(ithprime(n)) ;
end proc: # R. J. Mathar, Jul 23 2015 reusing code from A050412

Length@ NestWhileList[2 # + 1 &, Prime@ #, CompositeQ, {2, 1}] - 1 & /@ Range@ 120 (* Michael De Vlieger, Apr 26 2015 *)

genit()={
my(maxx=122,istrt=1,opt=1);n=istrt;cnt=1;val=2*prime(n)+1;
prev=val;prcnt=0;while(n<=maxx, if( val%6!=1 && val%6!=5,cnt+=1;val=2*val+1 );
if(ispseudoprime(val), print1(cnt,",");if(opt>0&&ispseudoprime(2*val+1),prcnt+=1);
cnt=1;n+=1;val=2*prime(n)+1;prev=val ); if(!ispseudoprime(val),cnt+=1;val=2*val+1));
}

a(n,k=prime(n))=my(t=1);while(!ispseudoprime(k=2*k+1),t++);t \\ Charles R Greathouse IV, May 22 2015

a(n) = A050412(prime(n)). - Michel Marcus, Jun 08 2015

A375155 a(n) = 2*a(n-1) + 1 for a(n-1) not prime, otherwise a(n) = prime(n) - 1; with a(1) = 147.

Original entry on oeis.org

147, 295, 591, 1183, 2367, 4735, 9471, 18943, 37887, 75775, 151551, 303103, 606207, 1212415, 2424831, 4849663, 9699327, 19398655, 38797311, 77594623, 155189247, 310378495, 620756991, 1241513983, 2483027967, 4966055935, 9932111871, 19864223743, 39728447487, 79456894975

Offset: 1

Chai Wah Wu, Aug 01 2024

nonn

Variant of A374965 with initial condition a(1) = 147. If a(1) is prime, the trajectory is: a(1), 2, 4, 9, ... and matches A374965 after the second term. It appears that for most composite a(1), the trajectories also converge towards A374965. This could be due to A050412(n) being small for many composite n.
a(1) = 147 is the first composite number where the trajectory appears not to converge towards A374965.
Other values of a(1) for which the trajectory converges to this sequence are: 295, 591, 1183, 2278, 2367, 3328, 4557, 4602, 4735, 5950, 6072, 6657, 7227, 9115, 9205, 9471, 9612, 9912, 9955, ...

Chai Wah Wu, Table of n, a(n) for n = 1..7697

Cf. A050412, A052333, A373799, A374965.

Module[{n = 1}, NestList[If[n++; PrimeQ[#], Prime[n] - 1, 2*# + 1] &, 147, 50]] (* Paolo Xausa, Aug 02 2024 *)

from itertools import islice
from sympy import isprime, nextprime
def A375155_gen(): # generator of terms
    a, p = 147, 3
    while True:
        yield a
        a, p = p-1 if isprime(a) else (a<<1)|1, nextprime(p)
A375155_list = list(islice(A375155_gen(),40))

A377364 a(n) = least k such that 2n*3^k-2 is prime, or 0 if no prime is reached.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 4, 5, 1, 2, 1, 2, 1, 1, 1, 9, 2, 1, 4, 1, 1, 2, 1, 5, 1, 1, 11, 1, 2, 2, 4, 3, 1, 1, 1, 3, 2, 4, 1, 1, 5, 3, 1, 1, 3, 1, 2, 4, 1, 1, 1, 2, 2, 1, 5, 1, 3, 1, 2, 1, 1, 8, 3, 1, 1, 4, 2, 80, 1, 6, 1, 8, 2, 2

Offset: 1

Clark Kimberling, Oct 31 2024

nonn

			a(20) = 5 because 40*3^5 + 1 is prime and 40*3^k + 1 is not prime for k=1..4.

Cf. A000040, A050412, A354747, A377365, A377366, A377367.

{b, h} = {3, 2}; f[n_, k_] := n*b^k - h
s[n_] := Select[Range[20], PrimeQ[f[n, #]] &, 1]
Flatten[Table[s[n], {n, 1, 200}]]

A377365 a(n) = least k such that 2n*5^k+1 is prime, or 0 if no prime is reached.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 4, 2, 1, 2, 1, 3, 8, 1, 1, 1036, 1, 3, 2, 1, 1, 2, 1, 1, 2, 4, 1, 2, 1, 3, 6, 2, 257, 2, 2, 1, 40, 1, 1, 4, 2, 1, 2, 10, 1, 4, 2, 1, 6, 1, 3, 2, 1, 15, 4, 1, 79, 48, 1, 1, 2, 1, 5, 6, 1, 1, 6, 4, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 6

Offset: 1

Clark Kimberling, Oct 31 2024

nonn

			a(20) = 1036 because 40*5^k+1 is prime for k=1036 and not prime for k=1..1035.

Cf. A000040, A050412, A354747, A377364, A377366, A377367.

f[n_, k_] := 2 n*5^k + 1;
s[n_] := Select[Range[5000], PrimeQ[f[n, #]] &, 1];
Flatten[Table[s[n], {n, 1, 500}]]

cp's OEIS Frontend

A050544 Numbers k such that 37*2^k-1 is prime.

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A078681 Start with n; iterate the process x -> ceiling(3x/2) until reach a prime. Sequence gives number of steps to reach a prime or 0 if no prime is ever reached.

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A138508 Semiprime analog of Riesel problem: start with n; repeatedly double and add 1 until reach a semiprime. Sequence gives number of steps to reach a semiprime or 0 if no semiprime is ever reached.

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A225721 Starting with x = n, the number of iterations of x := 2x - 1 until x is prime, or -1 if no prime exists.

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A257495 The number of iterations (x -> 2x+1) until a prime is found, starting with prime(n); or 0 if a prime is never found.

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A375155 a(n) = 2*a(n-1) + 1 for a(n-1) not prime, otherwise a(n) = prime(n) - 1; with a(1) = 147.

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A377364 a(n) = least k such that 2n*3^k-2 is prime, or 0 if no prime is reached.

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A377365 a(n) = least k such that 2n*5^k+1 is prime, or 0 if no prime is reached.

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