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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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..30.

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

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Last modified January 17 06:12 EST 2022. Contains 350378 sequences. (Running on oeis4.)