A271383 a(n) is the smallest k such that there are exactly n primes between k*(k-1) and k^2 and exactly n primes between k^2 and k*(k+1), or 0 if no such k exists.

2, 8, 13, 21, 32, 38, 46, 60, 85, 74, 102, 111

Offset: 1

Author

Felix Fröhlich, Apr 07 2016

Keywords

• nonn
• more

Comments

Does k exist for every n?

a(15)=135, a(17)=154. - Chai Wah Wu, Apr 16 2021

If they are nonzero, a(13)-a(14), a(16), a(18)-a(19), a(22), a(25)-a(26), a(33)-a(35) > 710000; a(20) = 200, a(21) = 192, a(23) = 258, a(24) = 277, a(27) = 317, a(28) = 283, a(29) = 314, a(30) = 352, a(31) = 365, a(32) = 461. - J.W.L. (Jan) Eerland, Dec 21 2022

A stronger statement can be made regarding the terms listed above that are either 0 or > 710000: No two primes < A002386(80) = 18361375334787046697 differ by more than A005250(80) = 1550, and sqrt(18361375334787046697) = 4285017541.946..., so for every k in the interval [710000, 4285017541] there are more than 458 primes in each of the two intervals [k*(k-1), k^2] and [k^2, k*(k+1)], so every term a(n) with n <= 35 that is either 0 or > 710000 is necessarily either 0 or > 4285017541 (with a(n) = 0 looking nearly certain). - Jon E. Schoenfield, Dec 23 2022

Examples

			For n = 6: 38*(38-1) = 1406, 38^2 = 1444 and 38*(38+1) = 1482. A000720(1444) - A000720(1406) = 6 and A000720(1482) - A000720(1444) = 6. Since 38 is the smallest k where the number of primes in both intervals is 6, a(6) = 38.
		

Links

• Wikipedia, Oppermann's conjecture.

Programs

Mathematica

Table[SelectFirst[Range[10^3], And[PrimePi[#^2] - PrimePi[# (# - 1)] == n, PrimePi[# (# + 1)] - PrimePi[#^2] == n] &], {n, 30}] /. k_ /; MissingQ@ k -> 0 (* Michael De Vlieger, Apr 09 2016, Version 10.2 *)

PARI

a(n) = my(k=1); while((primepi(k^2)-primepi(k*(k-1)))!=n || (primepi(k*(k+1))-primepi(k^2))!=n, k++); k

Extensions

Escape clause added to definition by Chai Wah Wu, Apr 17 2021