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A197504
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1 together with the odd numbers m >= 3 for which phi(2*m)/2 = phi(m)/2 is odd, where phi = A000010 (Euler's totient).
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5
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1, 3, 7, 9, 11, 19, 23, 27, 31, 43, 47, 49, 59, 67, 71, 79, 81, 83, 103, 107, 121, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 243, 251, 263, 271, 283, 307, 311, 331, 343, 347, 359, 361, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 529
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OFFSET
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1,2
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COMMENTS
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These are the odd numbers m for which the degree of the algebraic number sin(Pi/(2*m)) (the degree of its minimal polynomial), given by A055035(2*m), is odd. Because A055035(1) = 1, there is just this other instance with odd A055035.
This sequence {a(n)} consists for n >= 2 of all powers >= 1 of each prime p == 3 (mod 4) from A002145, sorted into increasing order. This follows from the factorization of odd m >= 3, and that phi(m)/2 has to be odd.
For a(n), with n >= 2, there is exactly one pair of solutions x = +1 and -1 (the trivial solution) of the congruence x^2 == +1 (mod a(n)), and there is no solution of the congruence x^2 == -1 (mod a(n)). The proof starts with showing this for p == 3 (mod 4). It can be shown that the square 1 appears only for x = 1 if x runs through 1, ..., (p-1)/2. The other x range (p+1)/2, ..., p-1, which has the same squares (mod p), can, by reading backwards, be interpreted as the -x partners of x from the first range. The Legendre symbol (-1, p) = -1 shows the second claim. Then one applies the lifting theorem for powers of primes (see Apostol, Theorem 5.30, p. 121), where only its part (a) is needed, and the step by step lifting to each prime power is unique.
For a(1) = 1 there is just one solution x = 0 of the congruence x^2 == +1 (mod 1), and also of x^2 == -1 (mod 1).
The complementary sequence with odd m >= 3 and even phi(m)/2 is given in A327922.
(End)
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 121-122.
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LINKS
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FORMULA
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a(1) = 1, and for n >= 2 the set of all positive powers of each prime p == 3 (mod 4) (A002145), sorted increasingly.
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EXAMPLE
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Factorization for n >= 2: 3, 7, 3^2, 11, 19, 23, 3^3, 31, 43, 47, 7^2, 59, 67, 71, 79, 3^4, 83, 103, 107, ... - Wolfdieter Lang, Nov 12 2019
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MATHEMATICA
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a[n_] := If[n == 2, 1, EulerPhi[n]/{1, 1, 2, 1}[[Mod[n, 4] + 1]]]; aa = {}; Do[If[OddQ[a[n]], AppendTo[aa, n/2]], {n, 2, 1000}]; aa
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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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EXTENSIONS
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STATUS
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approved
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