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A327922
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Odd numbers m >= 3 for which phi(2*m)/2 = phi(m)/2 is even, where phi = A000010 (Euler's totient).
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5
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5, 13, 15, 17, 21, 25, 29, 33, 35, 37, 39, 41, 45, 51, 53, 55, 57, 61, 63, 65, 69, 73, 75, 77, 85, 87, 89, 91, 93, 95, 97, 99, 101, 105, 109, 111, 113, 115, 117, 119, 123, 125, 129, 133, 135, 137, 141, 143, 145, 147, 149, 153, 155, 157, 159, 161, 165, 169, 171, 173, 175, 177, 181, 183, 185, 187, 189, 193
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OFFSET
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1,1
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COMMENTS
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This sequence is the complement of A197504 with respect to the positive odd numbers. It collects all positive odd numbers m with 4 dividing phi(m).
In the prime number factorization of an odd m >= 3 the largest even factor of phi(a(n)) is 2^{2*r1 + r3}, where r1 and r3 are the nonnegative number of distinct primes 1 (mod 4) and 3 (mod 4), respectively. This means that for m = a(n) one needs 2*r1 + r3 >= 2. See some examples below.
The number of solutions of the congruence x^2 == +1 (mod a(n)) or (inclusive) x^2 == -1 (mod a(n)) is 2^(r1 + r3) + delta_{r3,0}*2^r1, with 2*r1 + r3 >= 2, where r1 and r3 are the number of distinct primes 1 (mod 4) and 3 (mod 4), respectively, in the prime number factorization of a(n), and delta is the Kronecker symbol.
This follows from the result that primes 1 (mod 4) (A002144) have Legendre symbol (-1, p) = +1 and primes 3 (mod 4) (A002145) have (-1, p) = -1. The part (a) of the lifting theorem for powers of primes (Apostol, 5.30, pp. 121-122) is used. Also the theorem that for odd primes p there are exactly (p-1)/2 quadratic residues modulo p (and exactly (p-1)/2 nonresidues modulo p) is needed (see e.g., Silverman, ch. 20, Theorem 1, p. 151).
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 121-122.
J. H. Silverman, A Friendly Introduction to Number Theory, fourth ed., Pearson Education, Inc, 2014, ch. 20, pp. 149-155.
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LINKS
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FORMULA
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All members of the set {odd m >= 1: 4 | phi(m)} ordered increasingly.
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EXAMPLE
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[n, a(n), [r1, r3], number of solutions x (mod a(n)), [solutions]] (with pm for + or -):
[1, 5, [1, 0], 4, [pm1,pm2]],
[5, 21 = 3*7, [0, 2], 4, [pm1, pm8],
[20, 65 = 5*13, [1, 1], 8, [pm1, pm8, pm14, pm18]],
[34, 105 = 3*5*7, [1, 2], 8, [pm1, pm29, pm34, pm41].
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MATHEMATICA
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Select[Range[3, 200, 2], And[EvenQ[#1], #1 == #2] & @@ {EulerPhi[2 #]/2, EulerPhi[#]/2} &] (* Michael De Vlieger, Jun 28 2020 *)
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PROG
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(PARI) isok(m) = (m > 3) && (m % 2) && ((eulerphi(m) % 4) == 0); \\ Michel Marcus, Nov 13 2019
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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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