A345330 Composite numbers k for which m == 0 (mod k) is the only solution to m^(2^v(k-1)+1) == -m (mod k), where v(k) = A007814(k) is the 2-adic valuation of k.

9, 21, 25, 27, 33, 45, 49, 57, 63, 65, 69, 77, 81, 93, 99, 105, 117, 121, 125, 129, 133, 141, 145, 147, 161, 165, 169, 171, 177, 185, 189, 201, 207, 209, 213, 217, 225, 231, 237, 243, 245, 249, 253, 261, 265, 273, 279, 285, 289, 297, 301, 305, 309, 321, 325

Offset: 1

Author

Jianing Song, Jun 14 2021

Keywords

• nonn

Comments

For primes p, m^(2^v(p-1)+1) == -m (mod p) has only one solution m == 0 (mod p). This sequence gives that composite numbers that satisfy this condition.

All terms are odd since for even k, m == -1 (mod k) is a solution.

Odd composite k is a term if and only if v(p-1) <= v(k-1) for all prime factors p of k. Proof: Let k = (p_1)^(e_1)*(p_2)^(e_2)*...*(p_r)^(e_r) be an odd number. m^(2^v(k-1)+1) == -m (mod k) has only one solution if and only if m^(2^v(k-1)+1) == -m (mod (p_i)^(e_i)) has only one solution for 1 <= i <= r, or equivalently, m^(2^v(k-1)) == -1 (mod (p_i)^(e_i)) has no solution for 1 <= i <= r, or v(p_i-1) <= v(k-1) for 1 <= i <= r.

All prime powers of the form p^e for odd prime p and e >= 2 are terms. All Carmichael numbers (A002997) are also terms: if k is a Carmichael number, then p-1 | k-1 for all prime factors p.

Examples

			225 = 3^2 * 5^2 is a term since v(3-1) = 1 <= v(225-1) = 7, v(5-1) = 2 <= v(225-1) = 7. Also, the equation m^(2^v(225-1)+1) == -m (mod 225) has a unique solution m == 0 (mod 225).
1885 = 5 * 13 * 29 is a term since v(5-1) = v(13-1) = v(29-1) = 2 <= v(1885-1) = 2. Also, the equation m^(2^v(1885-1)+1) == -m (mod 1885) has a unique solution m == 0 (mod 1885).
		

Links

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

Crossrefs

Cf. A007814, A002997.

Complement of A345331 with respect to A071904.

Setwise difference A360114 \ A065091. Subsequence of A360117.

Programs

PARI

isA345330(n) = if(!isprime(n) && n>1 && n%2, my(f=factor(n), w=omega(n)); for(i=1, w, if(valuation(f[i,1]-1,2) > valuation(n-1,2), return(0))); 1, 0)

Extensions

Name revised by Michael B. Porter, Feb 22 2023