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Difference between prime(n)^2 and the previous prime.
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%I #25 Feb 27 2023 13:37:57

%S 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,

%T 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,

%U 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

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

%C From _Jean-Christophe Hervé_, Oct 22 2013: (Start)

%C Contains only even numbers, except the first term.

%C 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.

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

%H Harvey P. Dale, <a href="/A054271/b054271.txt">Table of n, a(n) for n = 1..1000</a>

%F 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]

%e From _Zak Seidov_, Feb 20 2012: (Start)

%e n=4 and prime(4)^2=49, preceded by prime(15)=47, so a(4)=49-47=2;

%e n=97 and prime(97)^2=509^2=259081, preceded by prime(22765)=259033, so a(97)=259081-259033=48. (End)

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

%o (PARI) a(n) = my(p=prime(n)); p^2 - precprime(p^2); \\ _Michel Marcus_, Feb 27 2023

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

%K nonn,easy

%O 1,2

%A _Labos Elemer_, May 05 2000