A352587 - OEIS

%I #39 Apr 24 2023 05:34:56

%S 2,4,6,10,16,18,20,28,60,84,228,240,280,366,420,468,484,604,684,942,

%T 990,1152,1170,1196,1440,2064,5292,5954,8968,9176,13242,13680,14160,

%U 15190,24524,28764,29422,30558,30646,34804,35190,38164,44642,56772,62790,93024

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

%C Any counterexample to the Goldbach conjecture must have this form.

%C 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).

%H Craig J. Beisel, <a href="/A352587/b352587.txt">Table of n, a(n) for n = 1..56</a>

%e 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.

%o (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,","); ); );

%Y Cf. A103131, A141098, A352612.

%K nonn

%O 1,1

%A _Craig J. Beisel_, Mar 21 2022