A108294 a(n) = the least prime p such that p-6n-3 is a power of 2 and 2p-6n-3 is prime, or -1 if no such prime exists.

5, 11, 17, 29, 29, 37, 41, 53, 59, 59, 67, 71, 79, 89

Offset

0,1

Comments

The value of a(14) is presently unknown.

A108295 has the 2p-6n-3 values.

From Robert Israel, Aug 04 2016: (Start)

If a(n) is not -1, it is 2^k+6n+3 where both 2^k+6n+3 and 2^(k+1)+6n+3 are prime, and k is the least number with this property.

Heuristically, the probability that 2^k+6n+3 is prime should be on the order of k^(-1), and the probability that both 2^k+6n+3 and 2^(k+1)+6n+3 are prime should be on the order of k^(-2). Thus we should expect only finitely many such k for a given n, and if it does not occur for fairly small k it is unlikely to ever occur. However, this is not a proof.

If it is not -1, a(14) > 2^10000+87. (End)

Values a(14) through a(60) for 2 <= p <= prime(10^6): {-1, 97, 101, 107, 239, 149, 127, 193, 137, 149, 149, 281, 163, 173, 32939, 179, 191, 191, 197, -1, 223, -1, 223, 227, 359, 239, -1, 281, 263, 269, 269, 277, 281, 317, 419, 809, 307, 311, 331, 337, -1, 397, -1, 347, 359, 373, 491}. Unknown values (-1) at n = 14, 33, 35, 40, 54, 56. - Michael De Vlieger, Aug 04 2016

Links

Table of n, a(n) for n=0..13.

Robert Israel, Table of n, (conjectured) a(n) for n=1..100. Values > 0 are correct, values of -1 are conjectured (with k > 10000 if a(n) > 0).

Maple

# values of -1 from this function indicate either a(n)=-1 or k > 10000
f:= proc(n) local j, k, p, q, state, goodk;
    state:= 0;
    goodk:= select(t -> igcd(6*n+3+2^t, 6*n+3+2^(t+1), 5*7*11*13*17)=1, [$0..119]);
    for k from 1 to 10000 do
        if (state = 0 and not member(k mod 120, goodk)) then state:= 0; next fi;
        p:= 6*n+3+2^k;
        if isprime(p) then
           state:= state+1;
           if state = 2 then return p - 2^(k-1) fi;
        else
           state:= 0;
        fi;
    od;
    -1;
end proc:
seq(f(n), n=0..30); # Robert Israel, Aug 04 2016

Mathematica

Table[SelectFirst[Prime@ Range[10^5], IntegerQ@ Log2[# - 6 n - 3] && PrimeQ[2 # - 6 n - 3] &], {n, 0, 60}] /. k_ /; MissingQ@ k -> -1 (* Michael De Vlieger, Aug 04 2016, Version 10.2 *)

Crossrefs

Cf. A108295.

Sequence in context: A089110 A023489 A268307 * A046869 A028388 A277718

Adjacent sequences: A108291 A108292 A108293 * A108295 A108296 A108297

Keyword

nonn,more

Author

Cino Hilliard, Jun 29 2005

Extensions

Edited by Don Reble, May 01 2007 and by N. J. A. Sloane, May 04 2007

Status

approved