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

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

1,1

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

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 = 1..1000

Example

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.

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=1..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 *)

Crossrefs

Cf. A005385, A006530, A051886, A074781.

Sequence in context: A195549 A294132 A125146 * A275800 A347475 A294136

Adjacent sequences: A051897 A051898 A051899 * A051901 A051902 A051903

Keyword

nonn

Author

Labos Elemer, Dec 16 1999

Extensions

Name clarified by Joerg Arndt, Jun 18 2012

Status

approved