A076957 Smallest k such that there are exactly n primes strictly between k^2 and (k+1)^2.

1, 4, 6, 10, 15, 16, 25, 24, 31, 39, 38, 45, 64, 48, 52, 57, 75, 82, 81, 70, 76, 79, 106, 112, 145, 111, 121, 117, 123, 134, 144, 139, 146, 154, 163, 192, 169, 176, 179, 193, 202, 218, 204, 226, 223, 240, 233, 238, 243, 259, 291, 256, 286, 309, 278

Offset: 2

Author

Amarnath Murthy, Oct 20 2002

Keywords

• nonn

Comments

From David W. Wilson, Jan 08 2017: (Start)

a(n)^2 = A076956(n).

A014085(a(n)) = n.

Conjecturally, a(n) is undefined for n = 1 and defined for all n >= 2. (End)

Links

• T. D. Noe and David W. Wilson, Table of n, a(n) for n = 2..10000 (a(n) for n = 2..1000 from T. D. Noe).

Crossrefs

Cf. A014085, A076956, A084597.

Programs

Maple

a := proc(n) local k, h, SEARCHLIMIT; SEARCHLIMIT := 10000; h := proc(k) option remember; nops(select(j->isprime(j), [$k^2+1..(k+1)^2])) end: k := 1; while h(k) <> n and k < SEARCHLIMIT do k := k+1 od; `if`(k=SEARCHLIMIT, print("Search limit reached!"), k) end: seq(a(n), n=2..56); # Peter Luschny, Jan 10 2017

Mathematica

Table[k = 1; While[Count[Map[PrimeQ, Range[k^2 + 1, (k + 1)^2]], True] != n, k++]; k, {n, 2, 56}] (* Michael De Vlieger, Jan 10 2017 *)
With[{pp=Table[Count[Range[n^2+1,(n+1)^2-1],?(PrimeQ[#]&)],{n,500}]},Table[ Position[pp,k,1,1],{k,60}]]//Flatten (* _Harvey P. Dale, Aug 01 2021 *)

Extensions

More terms from Ralf Stephan, Oct 31 2002