A218121 Numerator of c(n) defined by c(1)=1, c(2)=5/2 and for n>=3, c(n) is the minimal rational number >= c(n-1) such that there are no primes in the interval (Prime(n)/c(n), Prime(n+1)/c(n)).

1, 5, 5, 11, 11, 17, 17, 17, 29, 29, 29, 41, 41, 41, 41, 41, 41, 67, 67, 67, 67, 83, 83, 83, 83, 83, 83, 109, 109, 127, 127, 127, 127, 149, 149, 149, 149, 149, 149, 149, 181, 181, 181, 181, 181, 181, 181, 181, 229, 229, 229, 229, 251, 251, 251, 251, 251, 251, 251, 251

Offset

1,2

Comments

The sequence c(n) begins 1, 5/2, 5/2, 11/2, 11/2, 17/3, ...

Its terms > 1 are ratios of primes.

Links

Table of n, a(n) for n=1..60.

Formula

For n>=3, if interval (Prime(n)/c(n-1), Prime(n+1)/c(n-1)) is free from primes, then c(n)=c(n-1); otherwise, c(n)=Prime(n+1)/Prime(k), where k<=n is the maximal, such that a) Prime(n+1)/Prime(k)>c(n-1) and b) the open interval (Prime(n)*Prime(k)/Prime(n+1), Prime(k)) does not contain any prime.

Note that such k exists, since, for k=1, the interval (2*Prime(n)/Prime(n+1),2) is free from primes.

Example

Intervals (2/1,3/1),(3/(5/2),5/(5/2)) are free from primes. By the condition, c(3) >= c(2) = 5/2. Since also (5/(5/2),7/(5/2)) contains no prime, then c(3)=5/2. Further, c(4) should be chosen minimal>=5/2 such that the interval (7/c(4),11/c(4)) does not contain 2 and 3 (it is clear that it contains no prime>=5). It is easy to see that the minimal c(4)=11/2, etc.

Maple

ispfree := proc(a, b)
    local alow ;
    alow := floor(a);
    if nextprime(alow) < b then
        false;
    else
        true;
    end if;
end proc:
A218121c := proc(n)
    option remember;
    local k ;
    if n = 1 then
        return 1;
    elif n = 2 then
        return 5/2 ;
    else
        if ispfree(ithprime(n)/procname(n-1), ithprime(n+1)/procname(n-1)) then
            return procname(n-1) ;
        end if ;
        for k from n by -1 do
            if ispfree( ithprime(n)*ithprime(k)/ithprime(n+1), ithprime(k) )
                and ithprime(n+1)/ithprime(k) > procname(n-1) then
                return ithprime(n+1)/ithprime(k) ;
            end if;
        end do:
    end if;
end proc:
A218121 := proc(n)
    numer(A218121c(n)) ;
end proc: # R. J. Mathar, Dec 02 2012

Crossrefs

Cf. A218123, A217871, A217689, A217691, A217833, A217884.

Sequence in context: A143427 A287996 A239355 * A168300 A356048 A101203

Adjacent sequences: A218118 A218119 A218120 * A218122 A218123 A218124

Keyword

nonn,frac

Author

Vladimir Shevelev, Oct 21 2012

Status

approved