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A051900
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Minimal 2^n safe-primes: a(n) = 2^n*A051886(n) + 1 (a prime number).
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3
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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;
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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