A212037 The size of the set of numbers k>=0 such that all (2^n+k)*2^n-1 are prime but only the last (largest) (2^n+k)*2^n+1 is also an associated twin prime.

1, 6, 1, 4, 2, 2, 2, 24, 6, 2, 28, 7, 16, 47, 29, 6, 41, 16, 3, 17, 32, 10, 10, 23, 14, 15, 52, 4, 13, 20, 23, 4, 84, 26, 88, 50, 20, 35, 51, 44, 41, 87, 1, 142, 13, 188, 107, 162, 91, 96, 197, 4, 148, 71, 9, 66, 97, 41, 10, 9, 152, 234, 48, 104, 144, 40, 18, 45, 52, 204, 21, 49, 51, 9, 102, 13, 31, 108, 88

Offset

1,2

Comments

Starting at a count of zero, we consider for increasing k>=0 the pairs (2^n+k)*2^n+-1. If the smaller of these two numbers is prime, we increase the counter. If the larger of these two numbers is also prime, we admit the counter to the sequence. It is basically a measure of how many unsuccessful primality tests on the larger of the two numbers are done before it becomes a compatible twin prime.

Heuristically, the average of a(n)/n over n=1 to N tends to 1 as N increases.

Links

Pierre CAMI, Table of n, a(n) for n = 1..825

Example

For n=2, the 6 pairs (19,21) at k=1, (23,25) at k=2, (31,33) at k=4, (43,45) at k=7, (47,49) at k=8 and (59,61) at k=11 are counted. The smaller of these must be a prime to be counted, and at k=11 also the larger (i.e., 61) becomes prime, which finishes the search.

Maple

A212037 := proc(n)
    local a, k, p ;
    a := 0 ;
    for k from 0 do
        p := (2^n+k)*2^n-1 ;
        if isprime(p) then
            a := a+1 ;
        end if;
        if isprime(p) and isprime(p+2) then
            return a;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 20 2012

Prog

(PFGW & SCRIPT)
SCRIPT
DIM nn, 0
DIM jj
DIM kk
DIMS tt
OPENFILEOUT myfile, a(n).txt
LABEL loopn
SET nn, nn+1
IF nn>825 THEN END
SET kk, -1
SET jj, 0
LABEL loopk
SET kk, kk+1
SETS tt, %d, %d\,; nn; kk
PRP (2^nn+kk)*2^nn-1, tt
IF ISPRP THEN GOTO a
IF ISPRIME THEN GOTO a
GOTO loopk
LABEL a
SET jj, jj+1
PRP (2^nn+kk)*2^nn+1, tt
IF ISPRP THEN GOTO d
IF ISPRIME THEN GOTO d
GOTO loopk
LABEL d
SETS tt, %d, %d\,; nn; jj
WRITE myfile, tt
GOTO loopn

Crossrefs

Cf. A191617, A191218, A205321.

Sequence in context: A178646 A144540 A292107 * A118740 A301431 A200302

Adjacent sequences: A212034 A212035 A212036 * A212038 A212039 A212040

Keyword

nonn

Author

Pierre CAMI, Jul 14 2012

Status

approved