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A081858
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Numbers k such that 2*k+1 divides 2^k-1.
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7
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0, 3, 8, 11, 15, 20, 23, 35, 36, 39, 44, 48, 51, 56, 63, 68, 75, 83, 95, 96, 99, 111, 116, 119, 120, 128, 131, 135, 140, 155, 156, 168, 170, 176, 179, 183, 191, 200, 204, 215, 216, 219, 224, 228, 231, 239, 243, 251, 260, 280, 284, 288, 296, 299, 300, 303, 308
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OFFSET
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1,2
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COMMENTS
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n is a term if and only if n=0, 2n+1 is a prime of the form 8k+-1, or 2n+1 is an Euler pseudoprime satisfying 2^n == 1 mod 2n+1.
Case 1: 0 is a term. Case 2, 2n+1 prime: by Euler's criterion and the quadratic character of 2, 2^n == 1 mod 2n+1 only if 2n+1 is of the form 8k+-1. Case 3, 2n+1 composite: 2^n == 1 mod 2n+1 only if 2n+1 is one of the subset of Euler pseudoprimes satisfying 2^n == 1 mod 2n+1.
The first term for which 2n+1 is a qualifying Euler pseudoprime is n=170.
The first Euler pseudoprime that does not correspond to a term is 3277, because 2^((3277-1)/2) == -1 mod 3277. (End)
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LINKS
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FORMULA
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k such that A002326(k)|k: since 2^k == 1 mod 2*k+1, k must be a multiple of the order of 2 mod 2*k+1.
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MATHEMATICA
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Join[{0}, Select[Range[300], PowerMod[2, #, 2*# + 1] === 1 &]] (* Amiram Eldar, Jun 02 2022 *)
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PROG
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(PARI) isok(n) = !((2^n-1) % (2*n+1)); \\ Michel Marcus, Dec 04 2013
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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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