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A302254
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Exponent of the group of the Gaussian integers in a reduced system modulo (1+i)^n.
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2
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1, 1, 2, 4, 4, 4, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152, 4194304, 4194304, 8388608, 8388608
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OFFSET
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0,3
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COMMENTS
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For n > 0, the number of elements in the group of the Gaussian integers in a reduced system modulo (1+i)^n is 2^(n-1).
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LINKS
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FORMULA
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For n > 5, a(n) = 2^(floor(n/2) - 1).
For even n, a(n) = A227334(2^(n/2)).
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EXAMPLE
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For Gaussian integer x such that (x, 1+i) = 1, x^4 - 1 = (x + 1)(x - 1)(x + i)(x - i) provides at least 7 factors of 1+i in total (and exactly 7 when x = 2+i), so a(7) = 4.
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MATHEMATICA
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Join[{1, 1, 2, 4, 4, 4}, Table[2^(Floor[n/2] - 1), {n, 6, 50}]] (* Vincenzo Librandi, Apr 04 2018 *)
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PROG
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(Magma) [1, 1, 2, 4, 4, 4] cat [2^(Floor(n div 2)-1): n in [6..50]]; // Vincenzo Librandi, Apr 04 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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