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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.
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%I #23 Feb 26 2023 19:39:23

%S 9,21,25,27,33,45,49,57,63,65,69,77,81,93,99,105,117,121,125,129,133,

%T 141,145,147,161,165,169,171,177,185,189,201,207,209,213,217,225,231,

%U 237,243,245,249,253,261,265,273,279,285,289,297,301,305,309,321,325

%N 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.

%C 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.

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

%C 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.

%C 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.

%H Jianing Song, <a href="/A345330/b345330.txt">Table of n, a(n) for n = 1..10000</a>

%e 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).

%e 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).

%o (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)

%Y Cf. A007814, A002997.

%Y Complement of A345331 with respect to A071904.

%Y Setwise difference A360114 \ A065091. Subsequence of A360117.

%K nonn

%O 1,1

%A _Jianing Song_, Jun 14 2021

%E Name revised by _Michael B. Porter_, Feb 22 2023