A054271 Difference between prime(n)^2 and the previous prime.

1, 2, 2, 2, 8, 2, 6, 2, 6, 2, 8, 2, 12, 2, 2, 6, 12, 2, 6, 2, 6, 12, 6, 2, 6, 8, 2, 2, 14, 6, 2, 2, 12, 2, 8, 14, 18, 8, 6, 2, 12, 12, 2, 6, 6, 20, 2, 2, 8, 8, 2, 2, 8, 12, 2, 6, 8, 8, 12, 20, 12, 2, 20, 18, 2, 6, 14, 2, 8, 12, 8, 2, 6, 6, 12, 6, 18, 30, 12, 12, 18, 2, 8, 12, 24, 2, 2, 6, 14, 6

Offset

1,2

Comments

From Jean-Christophe Hervé, Oct 22 2013: (Start)

Contains only even numbers, except the first term.

Even integers of the form 3*k+1 (or equivalently integers of form 6*k+4) never appear because prime(n)^2 = 3*k+1 = 1 (mod 3), and prime(n)^2 - (3*k+1) is multiple of 3.

Conjecture: every other even integer appears in the sequence an infinite number of times. (End)

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

a(n) = prime(n)^2 - precprime(prime(n)^2), where precprime(x) is the largest prime less than x. [Corrected by Jean-Christophe Hervé, Oct 21 2013]

Example

From Zak Seidov, Feb 20 2012: (Start)
n=4 and prime(4)^2=49, preceded by prime(15)=47, so a(4)=49-47=2;
n=97 and prime(97)^2=509^2=259081, preceded by prime(22765)=259033, so a(97)=259081-259033=48. (End)

Mathematica

f[n_]:=Module[{n2=n^2}, n2-NextPrime[n2, -1]]; f/@Prime[Range[90]] (* Harvey P. Dale, Oct 19 2011 *)

Prog

(PARI) a(n) = my(p=prime(n)); p^2 - precprime(p^2); \\ Michel Marcus, Feb 27 2023

Crossrefs

Cf. A001248, A054270, A091666, A133517, A133518, A133519, A133520, A133521, A133522, A001223.

Sequence in context: A055921 A029605 A367189 * A278245 A321026 A240284

Adjacent sequences: A054268 A054269 A054270 * A054272 A054273 A054274

Keyword

nonn,easy

Author

Labos Elemer, May 05 2000

Status

approved