A367035 -id:A367035 - cp's OEIS frontend

A368549 a(n) is the least k such that the greatest prime less than n*k is less than n times the greatest prime < k.

8, 12, 24, 24, 20, 18, 38, 14, 54, 30, 138, 98, 152, 90, 84, 80, 240, 504, 68, 200, 354, 854, 224, 1020, 230, 180, 510, 542, 524, 522, 368, 968, 578, 1098, 1130, 3462, 744, 504, 1218, 1988, 468, 938, 812, 758, 684, 4002, 2592, 642, 3120, 4458, 2958, 3272, 4920, 572, 5060, 3300, 14490, 3188, 6012

Offset: 2

Robert Israel, Dec 29 2023

nonn

a(n) = p + 1 where p is the first prime such that there are no primes between n*p and n*(p+1).

			a(4) = 24 because the greatest prime < 24 is 23 and the greatest prime < 4*24 is 89 < 4*23.

Robert Israel, Table of n, a(n) for n = 2..280

Cf. A367035.

f:= proc(n) local p;
   p:= 1;
   do
     p:= nextprime(p);
     if not ormap(isprime, [$ (n*p+1) .. (n*p+n-1)]) then return p+1 fi
   od
end proc:
map(f, [$2..100]);

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

Original entry on oeis.org

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

Robert Israel, Dec 16 2023

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)

			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.

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

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);

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

cp's OEIS Frontend

A368549 a(n) is the least k such that the greatest prime less than n*k is less than n times the greatest prime < k.

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

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cp's OEIS Frontend

A368549 a(n) is the least k such that the greatest prime less than n*k is less than n times the greatest prime < k.

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Maple

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.

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Author

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Maple

Extensions

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