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1, 2, 4, 8, 16, 8, 36, 32, 36, 32, 100, 32, 144, 72, 64, 128, 256, 72, 324, 128, 144, 200, 484, 128, 400, 288, 324, 288, 784, 128, 900, 512, 400, 512, 576, 288, 1296, 648, 576, 512, 1600, 288, 1764, 800, 576, 968, 2116
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of split complex numbers z = x + yj in a reduced system modulo n where x, y are integers, j^2 = 1; number of solutions to gcd(x^2 - y^2, n)=1 with x, y in [0, n-1].
a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - 1) with discriminant d = 4, where Z_n is the ring of integers modulo n. (End)
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LINKS
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FORMULA
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Multiplicative with a(2^e) = 2^(2e-1) and a(p^e) = (p-1)^2*p^(2e-2) for p > 2. - R. J. Mathar, Apr 14 2011
a(n) = phi(n)^2 if n odd; 2*phi(n)^2 if n even, where phi(n) = A000010(n). - Jianing Song, Apr 20 2019
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/5) * Product_{p prime} (1 - (2*p-1)/p^3) = (2/5) * A065464 = 0.171299... . - Amiram Eldar, Oct 30 2022
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MAPLE
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A082953 := proc(n) numtheory[phi](n)*numtheory[phi](2*n) ; end proc:
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MATHEMATICA
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CROSSREFS
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Similar sequences: A127473 (size of (Z_n[x]/(x^2 - x))*, d = 1), A002618 ((Z_n[x]/(x^2))*, d = 0), A079458 ((Z_n[x]/(x^2 + 1))*, d = -4), A319445 ((Z_n[x]/(x^2 - x + 1))* or (Z_n[x]/(x^2 + x + 1))*, d = -3).
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KEYWORD
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mult,nonn
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AUTHOR
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Yuval Dekel (dekelyuval(AT)hotmail.com), May 26 2003
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STATUS
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approved
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