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A175625
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Numbers k such that gcd(k, 6) = 1, 2^(k-1) == 1 (mod k), and 2^(k-3) == 1 (mod (k-1)/2).
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4
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7, 11, 23, 31, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1123, 1187, 1283, 1291, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2543
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OFFSET
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1,1
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COMMENTS
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All composites in this sequence are 2-pseudoprimes, A001567. That subsequence begins with 536870911, 46912496118443, 192153584101141163, with no other composites below 2^64 (the first two were found by 'venco' from the dxdy.ru forum), and contains the terms of A303448 that are not multiples of 3. Correspondingly, composite terms include those of the form A007583(m) = (2^(2m+1) + 1)/3 for m in A303009. The only known composite member not of this form is a(1018243) = 536870911.
Intended as a pseudoprimality test; note that many primes do not pass the third condition either.
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LINKS
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MATHEMATICA
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Select[Array[(6 # + (-1)^# - 3)/2 &, 3000], And[PowerMod[2, (# - 1), #] == 1, PowerMod[2, (# - 3), (# - 1)/2] == 1] &] (* Michael De Vlieger, Dec 27 2023 *)
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PROG
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(PARI) isA175625(n) = gcd(n, 6)==1 && Mod(2, n)^(n-1)==1 && Mod(2, n\2)^(n-3)==1
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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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