A368208 a(n) is the least k such that, if p is the greatest prime less than k, there is a prime between n*p and n*k, but for 1 < j < n there is no prime between j*p and j*k.

3, 8, 32, 62, 138, 212, 464, 1610, 4458, 1952, 13004, 44742, 22778, 242814, 512718, 360198, 2366654, 1529030, 5532422, 13883834, 15516014, 51393768, 210568010, 271767438, 299891114, 758345724, 1204130100, 1363350560, 5171802930

Offset: 2

Author

Robert Israel, Dec 16 2023

Keywords

• nonn
• more

Comments

a(n) is the least k such that A049711(n*k) < n*A049711(k) but A049711(j*k) >= A049711(j*k) for 1 < j < n.

From David A. Corneth, Dec 17 2023: (Start)

An initial search for a(n) can be done over numbers one more than a prime i.e. of the form prime(m) + 1.

If a(n) is of the form prime(m) + u where there is no prime p between (exclusive) prime(m) and prime(m) + u then there is no prime between (n-1)*prime(m) and (n-1)*(prime(m) + u).

Looking at record gaps between primes in A002386 we need "pretty large" numbers for u > 1 so one could start searching with u = 1.

For 2 <= n <= 30 we have a(n) = prime(m) + 1 for some integer m. (End)

Examples

			a(4) = 32 because 31 is the greatest prime less than 32, and there are no primes between 2*31 = 62 and 2*32 = 64 and no primes between 3*31 = 93 and 3*32 = 96, but there is a prime between 4*31 = 124 and 4*32 = 128, namely 127.
		

Crossrefs

Cf. A002386, A008864, A049711, A367035, A151799.

Programs

Maple

f:= proc(n) local k,p;
  p:= prevprime(n);
  for k from 2 do
     if k*p < prevprime(k*n) then return k fi;
  od
end proc:
V:= Array(2..25): count:= 0:
for n from 3 while count < 24 do
  v:= f(n);
  if V[v] = 0 then V[v]:= n; count:= count+1 fi
od:
convert(V,list);

Extensions

a(28)..a(30) from David A. Corneth, Dec 17 2023