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%I #24 Oct 23 2023 01:44:06
%S 1,2,4,4,6,4,4,6,12,12,6,4,12,12,4,10,10,6,4,10,4,6,10,6,4,10,4,18,6,
%T 12,10,6,4,12,28,6,10,4,4,18,10,10,12,4,12,6,10,10,10,12,4,10,18,28,
%U 18,22,6,12,4,16,18,4,4,10,4,4,6,22,4,42,24,22,10,4
%N Difference between prime(n)^2 and the next prime.
%C Conjecturally, a(n) << log^2 n (with constant around 8/e^gamma in the supremum). [_Charles R Greathouse IV_, Dec 27 2011]
%C Except for a(2)=2, there are no terms = 2 mod 6 (as p^2+2 = 0 mod 3 for primes p > 3). Also, only 1 and 2 appear once while all other terms may appear (infinitely) many times. [_Zak Seidov_, Apr 18 2012]
%H Zak Seidov, <a href="/A091666/b091666.txt">Table of n, a(n) for n = 1..10000</a>
%F Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} log(prime(n))) = 2. - _Alain Rocchelli_, Oct 04 2023
%e prime(3)=5, 5*5=25 for k=4 25+4=29 prime, k=4 is the least k with prime(3)^2 + k prime.
%t NextPrime[#^2]-#^2&/@Prime[Range[74]] (* _Zak Seidov_, Apr 18 2012 *)
%o (PARI) a(n) = my(x=prime(n)^2); nextprime(x)-x; \\ _Michel Marcus_, Oct 07 2023
%Y Cf. A001248, A062772, A054271, A133517, A133518, A133519, A133520, A133521, A133522, A001223.
%K easy,nonn
%O 1,2
%A _Pierre CAMI_, Jan 27 2004
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