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.

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

Author

Jorge Coveiro, Jun 04 2005

Keywords

• nonn

Comments

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

References

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

Links

• 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.

Crossrefs

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

Cf. A040081, A046069.

Programs

Mathematica

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 *)

PARI

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

Extensions

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

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