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A074781
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Primes of the form p*2^k + 1 for any k and any prime p.
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11
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3, 5, 7, 11, 13, 17, 23, 29, 41, 47, 53, 59, 83, 89, 97, 107, 113, 137, 149, 167, 173, 179, 193, 227, 233, 257, 263, 269, 293, 317, 347, 353, 359, 383, 389, 449, 467, 479, 503, 509, 557, 563, 569, 587, 593, 641, 653, 719, 769, 773, 797, 809, 839, 857, 863, 887
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OFFSET
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1,1
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COMMENTS
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Equivalently, primes p such that the ratio (p-1)/gpf(p-1) = 2^k where gpf(m) is the greatest prime factor of m, A006530.
Paul Erdős asked if there are infinitely many primes p in this sequence (see R. K Guy reference). (End)
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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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3 = 2*2^0+1 is a term and 2/2 = 1 = 2^0.
7 = 3*2^1+1 is a term and 6/3 = 2 = 2^1.
13 = 3*2^2+1 is a term and 12/3 = 4 = 2^2.
41 = 5*2^3+1 is a term and 40/5 = 8 = 2^3.
113 = 7*2^4+1 is a term and 112/7 = 16 = 2^4.
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MAPLE
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alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2}:
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MATHEMATICA
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Select[Range[3, 1000], PrimeQ[#] && !CompositeQ[(# - 1)/2^IntegerExponent[(# - 1), 2]] &] (* Amiram Eldar, Dec 28 2018 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
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STATUS
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approved
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