A038699 -id:A038699 - cp's OEIS frontend

A194638 Smallest prime either of the form (2n-1)2^k - 1 or (2n-1)2^k + 1, k >= 0, or 0 if no such prime exists.

2, 2, 11, 13, 17, 23, 53, 29, 67, 37, 41, 47, 101, 53, 59, 61, 67, 71, 73, 79, 83, 173, 89, 751, 97, 101, 107, 109, 113, 1889, 487, 127, 131, 269, 137, 283, 293, 149, 307, 157, 163, 167, 1361, 173, 179, 181, 373, 191, 193, 197, 809, 823, 211, 857, 6977, 223, 227

Offset: 1

Arkadiusz Wesolowski, Aug 31 2011

nonn

Bisection of A194603.

Primes arising from A194636 (or 0 if no such prime exists).

			For n=4, 7*2^0-1 and 7*2^0+1 are composite, but 7*2^1-1=13 is prime, so a(4)=13.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194639.

Cf. A038699, A050921, A076335, A180247.

Table[n = 2*n - 1; k = 0; While[! PrimeQ[a = n*2^k - 1] && ! PrimeQ[a = n*2^k + 1], k++]; a, {n, 100}] (* Arkadiusz Wesolowski, Sep 04 2011 *)

A050921 Smallest prime of form n*2^m+1, m >= 0, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 7, 5, 11, 7, 29, 17, 19, 11, 23, 13, 53, 29, 31, 17, 137, 19, 1217, 41, 43, 23, 47, 97, 101, 53, 109, 29, 59, 31, 7937, 257, 67, 137, 71, 37, 149, 1217, 79, 41, 83, 43, 173, 89, 181, 47

Offset: 1

N. J. A. Sloane, Dec 30 1999

Primes arising from A040076 (or 0 if no such prime exists).

Or: Starting with x=n+1, the first prime created by iterating the map x-> 2*x-1. - Kevin L. Schwartz and Christian N. K. Anderson, May 13 2013

R. J. Mathar, Table of n, a(n) for n = 1..382

Cf. A040076, A034694, A038699.

A050921 := proc(n)
    for m from 0 do
        if isprime(n*2^m+1) then
            return n*2^m+1 ;
        end if;
    end do;
end proc; # R. J. Mathar, Jun 01 2013

Do[m = 0; While[ !PrimeQ[n*2^m + 1], m++ ]; Print[n*2^m + 1], {n, 1, 47} ]

The next term (47*2^583 + 1) is too large to show.

A108129 Riesel problem: let k=2n-1; then a(n)=smallest m >= 1 such that k*2^m-1 is prime, or -1 if no such prime exists.

Original entry on oeis.org

2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 4, 3, 1, 4, 1, 2, 2, 1, 3, 2, 7, 1, 4, 1, 1, 2, 1, 1, 12, 3, 2, 4, 5, 1, 2, 7, 1, 2, 1, 3, 2, 5, 1, 4, 1, 3, 2, 1, 1, 10, 3, 2, 10, 9, 2, 8, 1, 1, 12, 1, 2, 2, 25, 1, 2, 3, 1, 2, 1, 1, 2, 5, 1, 4, 5, 3, 2, 1, 1, 2, 3, 2, 4, 1, 2, 2, 1, 1, 8, 3, 4, 2, 1, 3, 226, 3, 1, 2, 1, 1, 2

Offset: 1

Jorge Coveiro, Jun 04 2005

nonn

It is conjectured that the integer k = 509203 is the smallest Riesel number, that is, the first n such that a(n) = -1 is 254602.
Browkin & Schinzel, having proved that 509203*2^k - 1 is composite for all k > 0, ask for the first such number with this property, noting that the question is implicit in Aigner 1961. - Charles R Greathouse IV, Jan 12 2018
Record values begin a(1) = 2, a(7) = 3, a(12) = 4, a(22) = 7, a(30) = 12, a(64) = 25, a(96) = 226, a(330) = 800516; the next record appears to be a(1147), unless a(1147) = -1. (The value for a(330), i.e., for k = 659, is from the Ballinger & Keller link, which also lists k = 2293, i.e., n = (k+1)/2 = (2293+1)/2 = 1147, as the smallest of 50 values of k < 509203 for which no prime of the form k*2^m-1 had yet been found.) - Jon E. Schoenfield, Jan 13 2018
Same as A046069 except for a(2) = 1. - Georg Fischer, Nov 03 2018

Hans Riesel, Några stora primtal, Elementa 39 (1956), pp. 258-260.

Jon E. Schoenfield, Table of n, a(n) for n = 1..329
A. Aigner, Folgen der Art ar^n + b, welche nur teilbare Zahlen liefern, Math. Nachr. 23 (1961), pp. 259-264. (Cited in Browkin & Schinzel)
R. Ballinger & W. Keller, The Riesel Problem: Definition and Status.
J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), pp. 55-58.
Wilfrid Keller, List of primes k.2^n - 1 for k < 300 .
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.

Main sequences for Riesel problem: A038699, A040081, A046069, A050412, A052333, A076337, A101036, A108129.

Cf. A040081, A046069.

Array[Function[k, SelectFirst[Range@300, PrimeQ[k 2^# - 1] &]][2 # - 1] &, 102] (* Michael De Vlieger, Jan 12 2018 *)
smk[n_]:=Module[{m=1,k=2n-1},While[!PrimeQ[k 2^m-1],m++];m]; Array[smk,120] (* Harvey P. Dale, Dec 26 2023 *)

forstep(k=1,301,2,n=1;while(!isprime(k*2^n-1),n++);print1(n,","))

Edited by Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 25 2006

Name corrected by T. D. Noe, Feb 13 2011

A103963 Record values in A040081.

Original entry on oeis.org

2, 3, 4, 7, 12, 23, 25, 2551, 800516

Offset: 1

Lei Zhou, Feb 24 2005

A040081(659) not yet found. The index sequence of this one is 1, 13, 23, 43, 59, 88, 127, 148, 659.

800516 was found by Dave Linton in 2004.

			A040081(1)=2; A040081(13)=3; A040081(659)>8192

Ray Ballinger and Wilfrid Keller, Riesel Problem

Cf. A038699, A040081, A240113.

kmax=8192; k = 0; n = 0; km = k; While[k < kmax, n++; k = 0; cp = n*(2^k) - 1; While[(!PrimeQ[cp]) && (k < kmax), k++; cp = n*(2^k) - 1]; If[k > km, km = k; Print[{n, km}]]]

Edited by T. D. Noe, Nov 15 2010

A057026 Smallest prime of form (2n+1)*2^m-1 for some m, or 0 if no such prime exists.

Original entry on oeis.org

3, 2, 19, 13, 17, 43, 103, 29, 67, 37, 41, 367, 199, 53, 463, 61, 131, 139, 73, 311, 163, 5503, 89, 751, 97, 101, 211, 109, 113, 241663, 487, 251, 1039, 2143, 137, 283, 9343, 149, 307, 157, 647, 331, 2719, 173, 1423, 181, 743, 379, 193, 197, 103423, 823, 419

Offset: 0

Henry Bottomley, Jul 24 2000

nonn

If a(329) > 0 it is greater than 659*2^10000. - Robert Israel, Jul 01 2014
Indeed, a(329) > 659*2^100000 if it is nonzero. There does not appear to be a covering set, though, so probably a(329) > 0. - Charles R Greathouse IV, Jul 02 2014
a(329) = 659*2^800516 - 1 (found by David W Linton in 2004). - Robert Israel, Jul 04 2014

			a(5)=43 because 2*5+1=11 and smallest prime of the form 11*2^m-1 is 43 (since 10 and 21 are not prime)

Cf. A057024, A057025, A038699.

A057026:= proc(n)
local t;
     t:= 2*n;
     while not isprime(t) do t:= 2*t+1 od;
     t
end proc;
seq(A057026(n),n=0..328); # Robert Israel, Jul 01 2014

A093007 First nonprime number reached when iterating n under x->2*x+1.

Original entry on oeis.org

1, 95, 15, 4, 95, 6, 15, 8, 9, 10, 95, 12, 27, 14, 15, 16, 35, 18, 39, 20, 21, 22, 95, 24, 25, 26, 27, 28, 119, 30, 63, 32, 33, 34, 35, 36, 75, 38, 39, 40, 335, 42, 87, 44, 45, 46, 95, 48, 49, 50, 51, 52, 215, 54, 55, 56, 57, 58, 119, 60, 123, 62, 63, 64, 65, 66, 135

Offset: 1

Reinhard Zumkeller, Mar 14 2004

nonn

			n = 41 = A000040(13) -> 2*41+1 = 83 = A000040(23) -> 2*83+1 = 167 = A000040(39) -> 2*167+1 = 335 = 67*5, therefore a(41) = 335, A063377(41) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A000120, A038699, A063377, A070939, A093009.

Table[NestWhile[2#+1&,n,PrimeQ],{n,70}] (* Harvey P. Dale, Sep 25 2012 *)

n>1: A070939(a(n)) = A070939(n) + A063377(n), A000120(a(n)) = A000120(n) + A063377(n).

Definition corrected by Harvey P. Dale, Sep 25 2012

A093009 Smallest number m such that m is prime iff n is not prime and the binary representation of m equals n with appended trailing 1's.

Original entry on oeis.org

3, 95, 15, 19, 95, 13, 15, 17, 19, 43, 95, 103, 27, 29, 31, 67, 35, 37, 39, 41, 43, 367, 95, 199, 103, 53, 223, 463, 119, 61, 63, 131, 67, 139, 71, 73, 75, 311, 79, 163, 335, 5503, 87, 89, 367, 751, 95, 97, 199, 101, 103, 211, 215, 109, 223, 113, 463, 241663

Offset: 1

Reinhard Zumkeller, Mar 14 2004

nonn

a(n) = Max{A093007(n), A038699(n)};

n>1: A070939(a(n))=A070939(n)+A093008(n), A000120(a(n))=A000120(n)+A093008(n).

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; }
}

A244609 Least prime divisor of 659*2^n-1.

Original entry on oeis.org

2, 3, 5, 3, 13, 3, 5, 3, 73, 3, 5, 3, 7, 3, 5, 3, 13, 3, 5, 3, 977, 3, 5, 3, 7, 3, 5, 3, 13, 3, 5, 3, 31, 3, 5, 3, 7, 3, 5, 3, 13, 3, 5, 3, 73, 3, 5, 3, 7, 3, 5, 3, 13, 3, 5, 3, 13477, 3, 5, 3, 7, 3, 5, 3, 13, 3, 5, 3, 48430237, 3, 5, 3, 7, 3, 5, 3, 13

Offset: 0

Robert Israel, Jul 01 2014

nonn

a(n) = 3 if n is odd.
a(n) = 5 if n == 2 (mod 4).
From Bruno Berselli, Jul 02 2014: (Start)
a(n) = 7 if n == 0 (mod 12) for n>0.
a(n) = 13 if n == 4 (mod 12).
a(n) == 3 or 7 (mod 12) for n>1. (End)
A040081(659) = 800516, so 800516 is the first n for which a(n) = 659*2^n-1 (found by David W Linton in 2004). - Jens Kruse Andersen, Jul 02 2014

			For n=4, 659*2^4-1 = 10543 = 13 * 811 so a(4) = 13.

Robert Israel, Table of n, a(n) for n = 0..355
The Prime Pages, 659*2^800516-1

Cf. A020639, A038699, A057026.

[PrimeDivisors(659*2^n-1)[1]: n in [0..100]]; // Bruno Berselli, Jul 02 2014

f:= proc(m) local F;
   F:= map(t -> t[1],ifactors(659*2^m-1,easy)[2]);
   F:= select(type,F,integer);
   if nops(F) = 0 then
     F:= map(t -> t[1],ifactors(659*2^m-1)[2]);
     min(F);
   else min(F)
   fi
end proc;
seq(f(n), n= 0 .. 100);

A178948 a(n) = smallest m>=0 such that n*(3^m)-1 is prime, or -1 if no such prime exists.

Original entry on oeis.org

1, 1, 0, 0, -1, 0, -1, 0, -1, 1, -1, 0, -1, 0, -1, 1, -1, 0, -1, 0, -1, 2, -1, 0, -1, 2, -1, 1, -1, 0, -1, 0, -1, 1, -1, 1, -1, 0, -1, 2, -1, 0, -1, 0, -1, 1, -1, 0, -1, 1, -1, 2, -1, 0, -1, 1, -1, 1, -1, 0, -1, 0, -1, 1, -1, 1, -1, 0, -1, 3, -1, 0, -1, 0, -1, 1, -1, 1, -1, 0, -1, 3, -1, 0, -1, 1, -1, 1, -1, 0

Offset: 1

Jonathan Vos Post, Dec 31 2010

sign

For odd n>=5, a(n)=-1 because then n*3^m is odd, therefore n*3^m-1 even and not prime. - R. J. Mathar, Jan 03 2011

			a(1) = 1 because 1*(3^1) - 1 = 2 is prime.
a(5) = -1 because 5*(3^m) - 1 is even and >=4.
a(22) = 2 because 22*(3^2)-1 = 197 is prime.

Cf. A038699, A050921.

cp's OEIS Frontend

A194638 Smallest prime either of the form (2*n-1)*2^k - 1 or (2*n-1)*2^k + 1, k >= 0, or 0 if no such prime exists.

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A050921 Smallest prime of form n*2^m+1, m >= 0, or 0 if no such prime exists.

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A108129 Riesel problem: let k=2n-1; then a(n)=smallest m >= 1 such that k*2^m-1 is prime, or -1 if no such prime exists.

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A103963 Record values in A040081.

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A057026 Smallest prime of form (2n+1)*2^m-1 for some m, or 0 if no such prime exists.

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Maple

A093007 First nonprime number reached when iterating n under x->2*x+1.

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A093009 Smallest number m such that m is prime iff n is not prime and the binary representation of m equals n with appended trailing 1's.

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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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A194638 Smallest prime either of the form (2n-1)2^k - 1 or (2n-1)2^k + 1, k >= 0, or 0 if no such prime exists.