1,2

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

Its terms > 1 are ratios of primes.

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.

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.

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

ispfree[a_, b_] := NextPrime[Floor[a]] >= b;

c[n_] := c[n] = Module[{k}, Which[n == 1, Return[1], n == 2, Return[5/2], True, If[ispfree[Prime[n]/c[n-1], Prime[n+1]/c[n-1]], Return[c[n-1]]]; For[k = n, True, k--, If[ispfree[Prime[n]*Prime[k]/Prime[n+1], Prime[k]] && Prime[n+1]/Prime[k] > c[n-1], Return[Prime[n+1]/Prime[k]]]]]];

a[n_] := Numerator[c[n]];

Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 01 2023, after R. J. Mathar *)

Cf. A218123 (denominator), A217871, A217689, A217691, A217833, A217884.

Sequence in context: A143427 A287996 A239355 * A168300 A356048 A101203

Adjacent sequences: A218118 A218119 A218120 * A218122 A218123 A218124

nonn,frac

Vladimir Shevelev, Oct 21 2012

approved