A152461 Primes p such that there does not exist any positive integer k and prime q with p > q and 3^k = p + 2q or 3^k = q + 2p.

2, 7, 19, 41, 53, 61, 73, 79, 83, 89, 127, 131, 139, 151, 163, 167, 173, 179, 191, 193, 199, 211, 223, 227, 241, 257, 277, 293, 317, 337, 373, 379, 389, 397, 401, 409, 419, 421, 433, 439, 443, 449, 457, 461, 463, 479, 487, 491, 499

Offset: 1

Author

Vladimir Shevelev, Dec 05 2008, Dec 12 2008

Keywords

• nonn

Comments

Powers of 3 are not expressible by sums of the form p + 2q, where p, q are terms of this sequence.

If there exists a sequence N_k = 3^n_k such that N_k has O((N_k)^v), v < 1/2, representations of the considered form, then removing the maximal primes in every such representation, we obtain an analog B of A152461 with the counting function Z(x) = pi(x) - O(x^v). Replacing in B the first N terms with N consecutive primes (with arbitrarily large N), we obtain a sequence which essentially is indistinguishable from the sequence of all primes with the help of the approximation of pi(x) by li(x), since, according to the well-known Littlewood result, the remainder term in the theorem of primes cannot be less than sqrt(x)logloglog(x)/log(x). But for this sequence we have infinitely many odd numbers which are not expressible by sum p + 2q with p, q primes. Thus in this case the Lemoine-Levy conjecture is essentially unprovable. Nevertheless, we conjecture that there does not exist a considered abnormal case of sequence (N_k).

Crossrefs

Cf. A152460 (complement).

Formula

If A(X) is the counting function of the terms a(n) <= x, then A(x) = x/log(x) + O(x*log(log(x))/(log(x))^2).