A161846 Numerator of the ratio (prime((n+1)^2) - prime(n^2))/prime(n).

5, 16, 6, 44, 54, 76, 84, 108, 122, 120, 166, 182, 184, 234, 192, 260, 264, 294, 304, 342, 378, 342, 408, 426, 414, 468, 488, 474, 516, 576, 588, 576, 604, 590, 696, 694, 728, 694, 756, 828, 774, 776, 870, 862, 852, 1010, 922, 998, 916, 1020, 1032, 1110, 1104

Offset: 1

Author

Daniel Tisdale, Jun 20 2009

Keywords

• nonn
• frac

Comments

Note that prime(n^2) = A011757(n) and prime(n) = A000040(n).

Conjecture: the sequence of fractions (prime((n+1)^2) - prime(n^2)) / prime(n) converges to 4. There are several "heuristic demonstrations" but no proofs.

Examples

			The first few fractions are 5/2, 16/3, 6/1, 44/7, 54/11, ...= A161846/A161847.
		

Crossrefs

Cf. A000040, A011757, A161847 (denominators).

Programs

Maple

A161846 := proc(n) ( ithprime((n+1)^2)-ithprime(n^2))/ithprime(n) ; numer(%) ; end: seq(A161846(n),n=1..25) ; # R. J. Mathar, Jun 22 2009

Mathematica

Table[(Prime[(n+1)^2]-Prime[n^2])/Prime[n],{n,60}]//Numerator (* Harvey P. Dale, Oct 24 2017 *)

PARI

a(n) = numerator((prime((n+1)^2) - prime(n^2))/prime(n)); \\ Michel Marcus, May 14 2020

Formula

a(n) = numerator((A011757(n+1) - A011757(n))/A000040(n)). - Petros Hadjicostas, May 13 2020

Extensions

Extended by Ray Chandler, May 06 2010

Various sections edited by Petros Hadjicostas, May 13 2020