A051900 Minimal 2^n safe-primes: a(n) = 2^n*A051886(n) + 1 (a prime number).

3, 5, 13, 17, 113, 97, 193, 257, 769, 11777, 13313, 59393, 12289, 40961, 114689, 65537, 2424833, 6946817, 786433, 5767169, 7340033, 23068673, 155189249, 595591169, 1224736769, 167772161, 469762049, 2281701377, 3489660929, 12348030977, 3221225473

Offset: 0

Author

Labos Elemer, Dec 16 1999

Keywords

• nonn

Comments

Equivalently, a(n) is the smallest prime p such that (p-1)/gpf(p-1) = 2^n where gpf(m) is the greatest prime factor of m, A006530. Subsequence of A074781, primes p such that the ratio (p-1)/gpf(p-1) = 2^k. - Bernard Schott, Dec 14 2020

Examples

			1 + 2^11*A051886(11) = 2048*29 + 1 = 59393 = a(11) is the smallest q prime so that (q-1)/2048 is also a (minimal, generalized Germain-) prime. The 101st term is 2385718429629527733616795432517633 = 1 + (2^101)*941.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A005385, A006530, A051886, A074781.

Programs

Maple

alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
a := proc(n) local p, q; q := 2^n; p := 2;
   while (p-1) <> gpf(p-1)*q
   do p := nextprime(p) od;
p end: seq(a(n), n=0..14); # Peter Luschny, Dec 14 2020

Mathematica

f[n_] := Block[{e = IntegerExponent[n - 1, 2]}, g = (n - 1)/2^e; If[g == 1, e - 1, If[ PrimeQ[g], e, -1]]]; t = Table[0, {50}]; p = 3; While[p < 13000000000, a = f@ p; If[t[[a + 1]] == 0, t[[a + 1]] = p; Print[{a, p}]]; p = NextPrime@ p]; t  (* Robert G. Wilson v, Jun 17 2012 *)
f[n_] := Block[{k = 1}, While[ !PrimeQ[2^n*Prime[k] + 1], k++]; 2^n*Prime[k] + 1]; Array[f, 32, 0] (* Robert G. Wilson v, Jun 17 2012 *)

Extensions

Name clarified by Joerg Arndt, Jun 18 2012

Offset changed to 0 and a(0) prepended by Amiram Eldar, Feb 28 2025