A319100 Number of solutions to x^6 == 1 (mod n).

1, 1, 2, 2, 2, 2, 6, 4, 6, 2, 2, 4, 6, 6, 4, 4, 2, 6, 6, 4, 12, 2, 2, 8, 2, 6, 6, 12, 2, 4, 6, 4, 4, 2, 12, 12, 6, 6, 12, 8, 2, 12, 6, 4, 12, 2, 2, 8, 6, 2, 4, 12, 2, 6, 4, 24, 12, 2, 2, 8, 6, 6, 36, 4, 12, 4, 6, 4, 4, 12, 2, 24, 6, 6, 4, 12, 12, 12, 6, 8, 6, 2

Offset

1,3

Comments

All terms are 3-smooth. a(n) is even for n > 2. Those n such that a(n) = 2 are in A066501.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Steven Finch, Greg Martin and Pascal Sebah, Roots of unity and nullity modulo n, Proc. Amer. Math. Soc., Vol. 138, No. 8 (2010), pp. 2729-2743.

Formula

Multiplicative with a(2) = 1, a(4) = 2, a(2^e) = 4 for e >= 3; a(3) = 2, a(3^e) = 6 if e >= 2; for other primes p, a(p^e) = 6 if p == 1 (mod 6), a(p^e) = 2 if p == 5 (mod 6).

If the multiplicative group of integers modulo n is isomorphic to C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j; then a(n) = Product_{i=1..m} gcd(6, k_i).

a(n) = A060594(n)*A060839(n).

For n > 2, a(n) = A060839(n)*2^A046072(n).

a(n) = A060594(n) iff n is not divisible by 9 and no prime factor of n is congruent to 1 mod 6, that is, n in A088232.

a(n) = A000010(n)/A293483(n). - Jianing Song, Nov 10 2019

Sum_{k=1..n} a(k) ~ c * n * log(n)^3, where c = (1/Pi^4) * Product_{p prime == 1 (mod 6)} (1 - (12*p-4)/(p+1)^3) = 0.0075925601... (Finch et al., 2010). - Amiram Eldar, Mar 26 2021

Example

Solutions to x^6 == 1 (mod 13): x == 1, 3, 4, 9, 10, 12 (mod 13).
Solutions to x^6 == 1 (mod 27): x == 1, 8, 10, 17, 19, 26 (mod 27) (x == 1, 8 (mod 9)).
Solutions to x^6 == 1 (mod 37): x == 1, 10, 11, 26, 27, 36 (mod 37).

Prog

(PARI) a(n)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(6, Z[i]))

Crossrefs

Number of solutions to x^k == 1 (mod n): A060594 (k=2), A060839 (k=3), A073103 (k=4), A319099 (k=5), this sequence (k=6), A319101 (k=7), A247257 (k=8).

Cf. A046072, A066501, A088232, A293483, A000010.

Mobius transform gives A307381.

Sequence in context: A355192 A260983 A103222 * A304794 A175809 A061033

Adjacent sequences: A319097 A319098 A319099 * A319101 A319102 A319103

Keyword

nonn,easy,mult

Author

Jianing Song, Sep 10 2018

Status

approved