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

Author

Labos Elemer, May 05 2000

Keywords

• nonn
• easy

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)

Examples

			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)
		

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

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]