A352587 Even numbers 2m such that A352612(2m) = A103131(2m).

2, 4, 6, 10, 16, 18, 20, 28, 60, 84, 228, 240, 280, 366, 420, 468, 484, 604, 684, 942, 990, 1152, 1170, 1196, 1440, 2064, 5292, 5954, 8968, 9176, 13242, 13680, 14160, 15190, 24524, 28764, 29422, 30558, 30646, 34804, 35190, 38164, 44642, 56772, 62790, 93024

Offset: 1

Author

Craig J. Beisel, Mar 21 2022

Keywords

• nonn

Comments

Any counterexample to the Goldbach conjecture must have this form.

Conjecture: For all a(n) > 18, a(n) is never equal to 2*q^x where q is prime and x is an integer x > 0. In other words, the product of its totatives is never congruent to -1 (mod 2m).

Examples

			For a(1) we have A352612(228) == -(59)(85) (mod 228) == 1 (mod 228) == A103131(228). Therefore A352612(228) == A103131(228) and 228 belongs to the sequence.
		

Links

• Craig J. Beisel, Table of n, a(n) for n = 1..56

Crossrefs

Cf. A103131, A141098, A352612.

Programs

PARI

for(n=1,150000, prod_t=1; prod_p=1; prod_r=1; for(k=3, 2*n-3, if(gcd(k,2*n)==1, prod_t=prod_t*k; ); if(gcd(k,2*n)==1 && isprime(k), prod_p=prod_p*k*(2*n-k); ); if(gcd(k,2*n)==1 && !isprime(k) && !isprime(2*n-k), prod_r=prod_r*k; ); ); if(-prod_t%(2*n)==(-prod_p*prod_r)%(2*n), print1(2*n,","); ); );