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A075432 Primes with no squarefree neighbors. 14

%I #43 Aug 13 2023 07:04:59

%S 17,19,53,89,97,127,149,151,163,197,199,233,241,251,269,271,293,307,

%T 337,349,379,449,487,491,521,523,557,577,593,631,701,727,739,751,773,

%U 809,811,881,883,919,953,991,1013,1049,1051,1061,1063,1097,1151,1171,1249

%N Primes with no squarefree neighbors.

%C Complement of A075430 in A000040.

%C From Ludovicus (luiroto(AT)yahoo.com), Dec 07 2009: (Start)

%C I propose a shorter name: non-Euclidean primes. That is justified by the Euclid's demonstration of the infinitude of primes. It appears that the proportion of non-Euclidean primes respect to primes tend to the limit 1-2A where A = 0.37395581... is Artin's constant. This table calculated by Jens K. Andersen corroborates it:

%C 10^5: 2421 / 9592 = 0.2523978315

%C 10^6: 19812 / 78498 = 0.2523885958

%C 10^7: 167489 / 664579 = 0.2520227091

%C 10^8: 1452678 / 5761455 = 0.2521373507

%C 10^9: 12817966 / 50847534 = 0.2520862860

%C 10^10: 114713084 / 455052511 = 0.2520875750

%C 10^11: 1038117249 / 4118054813 = 0.2520892256

%C It comes close to the expected 1-2A. (End)

%C This sequence is infinite by Dirichlet's theorem, since there are infinitely many primes == 17 or 19 (mod 36) and these have no squarefree neighbors. Ludovicus's conjecture about density is correct. Capsule proof: either p-1 or p+1 is divisible by 4, so it suffices to consider the other number (without loss of generality, p+1). For some fixed bound L, p is not divisible by any prime q < L (with finitely many exceptions) so there are q^2 - q possible residue classes for p. The primes in each are uniformly distributed so the probability that p+1 is divisible by q^2 is 1/(q^2 - q). The product of the complements goes to 2A as L increases without bound, and since 2A is an upper bound the limit is sandwiched between. - _Charles R Greathouse IV_, Aug 27 2014

%H Reinhard Zumkeller, <a href="/A075432/b075432.txt">Table of n, a(n) for n = 1..10000</a>

%H Pieter Moree, <a href="http://arxiv.org/abs/math/0412262">Artin's primitive root conjecture -a survey -</a>, arXiv:math/0412262 [math.NT], 2004-2012.

%H Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_065.htm">Conjecture 65. Non-Euclidean primes</a>, The Prime Puzzles and Problems Connection.

%F a(n) ~ Cn log n, where C = 1/(1 - 2A) = 1/(1 - Product_{p>2 prime} (1 - 1/(p^2-p))), where A is the constant in A005596. - _Charles R Greathouse IV_, Aug 27 2014

%p filter:= n -> isprime(n) and not numtheory:-issqrfree(n+1) and not numtheory:-issqrfree(n-1):

%p select(filter, [seq(2*i+1, i=1..1000)]); # _Robert Israel_, Aug 27 2014

%t lst={}; Do[p=Prime[n]; If[ !SquareFreeQ[Floor[p-1]] && !SquareFreeQ[Floor[p+1]], AppendTo[lst,p]], {n,6!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 20 2008 *)

%t Select[Prime[Range[300]],!SquareFreeQ[#-1]&&!SquareFreeQ[#+1]&] (* _Harvey P. Dale_, Apr 24 2014 *)

%o (Haskell)

%o a075432 n = a075432_list !! (n-1)

%o a075432_list = f [2, 4 ..] where

%o f (u:vs@(v:ws)) | a008966 v == 1 = f ws

%o | a008966 u == 1 = f vs

%o | a010051' (u + 1) == 0 = f vs

%o | otherwise = (u + 1) : f vs

%o -- _Reinhard Zumkeller_, May 04 2013

%o (PARI) is(n)=!issquarefree(if(n%4==1, n+1, n-1)) && isprime(n) \\ _Charles R Greathouse IV_, Aug 27 2014

%Y Cf. A039787, A049097, A005117, A000040, A008966, A010051.

%Y Intersection of A000040 and A281192.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Sep 15 2002

%E More terms (that were already in the b-file) from _Jeppe Stig Nielsen_, Apr 23 2020

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Last modified March 28 13:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)