OFFSET
1,2
COMMENTS
The primes are 3, 5 and those given in A191018 (Jacobi(prime|15) = +1).
For a(n) neither a multiple of 3 nor of 5 the Jacobi(a(n)|15) = +1.
The sequence (a(n))_{n >=1} is the set S := {3^a*5^b*Product_{j=1..m} (p_j)^{e(j)}}, in increasing order, with a and b from {0, 1}, primes p_j from A191018, m >= 0, and the exponents e(j) >= 0. If a = b = m = 0 then S = {1} and a(1) = 1.
The multiplicity of the (representative) solutions x is 2^m(n) for modulus a(n) from the set S. Thus it is 1 for 1, 3, 5, and a power of 2 with m(n) >= 1. This follows from Jacobi(prime|15) = +1, and the lifting theorem for powers of these primes (see e.g., Apostol). Primes 3 and 5 have only 1 solution and no lifting to powers >= 2 is possible. See A347833 for these multiplicities.
For the solutions x see A347832.
REFERENCES
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp 121, 122.
PROG
(PARI) isok(m) = {my(f=factor(m)); for (k=1, #f~, my(p=f[k, 1]); if ((p==3) || (p==5), if (f[k, 2] > 1, return (0)), if (kronecker(p, 15) != 1, return(0))); ); return (1); } \\ Michel Marcus, Oct 23 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 15 2021
STATUS
approved