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A014657
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Numbers m that divide 2^k + 1 for some nonnegative k.
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12
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1, 2, 3, 5, 9, 11, 13, 17, 19, 25, 27, 29, 33, 37, 41, 43, 53, 57, 59, 61, 65, 67, 81, 83, 97, 99, 101, 107, 109, 113, 121, 125, 129, 131, 137, 139, 145, 149, 157, 163, 169, 171, 173, 177, 179, 181, 185, 193, 197, 201, 205, 209, 211, 227, 229, 241, 243, 249, 251, 257, 265
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OFFSET
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1,2
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COMMENTS
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Since for some a < n, 2^a == 1 (mod n) (a consequence of Euler's Theorem), searching up to k=n is sufficient to determine whether an integer is in the sequence. - Michael B. Porter, Dec 06 2009
This sequence is the subset of odd integers > 1 as (2*n - 1) in A179480, such that the corresponding entry in A179480 is odd. Example: A179480(14) = 5, odd, with (2*14 - 1) = 27; and 5 is a term of this sequence. A014659 (odd and does not divide (2^k + 1) for any k >= 1) represents the subset of odd terms >1 corresponding to A179480 entries that are even. - Gary W. Adamson, Aug 20 2012
All prime factors of a(n) are in A091317. Sequence has asymptotic density 0. - Robert Israel, Aug 12 2014
This sequence, for m>2, is those m for which, for some e, (m-1)(2^e-1)/m is a term of A253608. Moreover, e(n) is 2*A195610(n) when m is a(n). - Donald M Davis, Jan 12 2018
Without a(2) = 2 this is the complement of A014659 relative to the odd positive integers A005408.
For the least nonnegative integer k(n) with 2^k(n) + 1 == d(n)*a(n), for n >= 1, see k(n) = A195610(n) and d(n) = A337220(n).
Starting with a(3) = 3 these numbers are the odd moduli, named 2*n+1 in the definition of A003558, for which the minus signs applies (see A332433(m) for the signs applying for A003558(m)). (End)
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LINKS
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MAPLE
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select(t -> [msolve(2^x+1, t)] <> [], [2*i+1 $ i=1..1000]); # Robert Israel, Aug 12 2014
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MATHEMATICA
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ok[n_] := Module[{k=0}, While[k<=n && Mod[2^k + 1, n] > 0, k++]; k<n]; Select[Range[265], ok] (* Jean-François Alcover, Apr 06 2011, after PARI prog *)
okQ[n_] := Module[{k = MultiplicativeOrder[2, n]}, EvenQ[k] && Mod[2^(k/2) + 1, n] == 0]; Join[{1, 2}, Select[Range[3, 265, 2], okQ]] (* T. D. Noe, Apr 06 2011 *)
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PROG
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(PARI) isA014657(n) = {local(r); r=0; for(k=0, n, if(Mod(2^k+1, n)==Mod(0, n), r=1)); r} \\ Michael B. Porter, Dec 06 2009
(Haskell)
import Data.List (findIndices)
a014657 n = a014657_list !! (n-1)
a014657_list = map (+ 1) $ findIndices (> 0) $ map a195470 [1..]
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CROSSREFS
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Besides initial terms 1 and 2, a subsequence of A296243. Their set difference is given by A296244.
Cf. A000051, A003558, A005408, A014659, A014661, A091317, A179480, A195470, A195610, A332433, A337220.
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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