cp's OEIS Frontend

A075432 Primes with no squarefree neighbors.

%I A075432 #51 Aug 03 2025 04:39:42
%S A075432 17,19,53,89,97,127,149,151,163,197,199,233,241,251,269,271,293,307,
%T A075432 337,349,379,449,487,491,521,523,557,577,593,631,701,727,739,751,773,
%U A075432 809,811,881,883,919,953,991,1013,1049,1051,1061,1063,1097,1151,1171,1249
%N A075432 Primes with no squarefree neighbors.
%C A075432 Complement of A075430 in A000040.
%C A075432 From Ludovicus (luiroto(AT)yahoo.com), Dec 07 2009: (Start)
%C A075432 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 A075432      10^5:        2421 /       9592 = 0.2523978315
%C A075432      10^6:       19812 /      78498 = 0.2523885958
%C A075432      10^7:      167489 /     664579 = 0.2520227091
%C A075432      10^8:     1452678 /    5761455 = 0.2521373507
%C A075432      10^9:    12817966 /   50847534 = 0.2520862860
%C A075432      10^10:  114713084 /  455052511 = 0.2520875750
%C A075432      10^11: 1038117249 / 4118054813 = 0.2520892256
%C A075432 It comes close to the expected 1-2A. (End)
%C A075432 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
%C A075432 Primes p such that both p-1 and p+1 are divisible by a square greater than 1. - _N. J. A. Sloane_, Jul 19 2024
%H A075432 Reinhard Zumkeller, <a href="/A075432/b075432.txt">Table of n, a(n) for n = 1..10000</a>
%H A075432 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 A075432 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 A075432 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
%e A075432 p = 17 is a term because 16 = 4^2 and 18=2*3^2 are divisible by squares > 1. - _N. J. A. Sloane_, Jul 19 2024
%p A075432 filter:= n -> isprime(n) and not numtheory:-issqrfree(n+1) and not numtheory:-issqrfree(n-1):
%p A075432 select(filter, [seq(2*i+1, i=1..1000)]); # _Robert Israel_, Aug 27 2014
%t A075432 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 A075432 Select[Prime[Range[300]],!SquareFreeQ[#-1]&&!SquareFreeQ[#+1]&] (* _Harvey P. Dale_, Apr 24 2014 *)
%o A075432 (Haskell)
%o A075432 a075432 n = a075432_list !! (n-1)
%o A075432 a075432_list = f [2, 4 ..] where
%o A075432    f (u:vs@(v:ws)) | a008966 v == 1 = f ws
%o A075432                    | a008966 u == 1 = f vs
%o A075432                    | a010051' (u + 1) == 0 = f vs
%o A075432                    | otherwise            = (u + 1) : f vs
%o A075432 -- _Reinhard Zumkeller_, May 04 2013
%o A075432 (PARI) is(n)=!issquarefree(if(n%4==1, n+1, n-1)) && isprime(n) \\ _Charles R Greathouse IV_, Aug 27 2014
%Y A075432 Cf. A039787, A049097, A005117, A000040, A008966, A010051.
%Y A075432 Intersection of A000040 and A281192.
%K A075432 nonn
%O A075432 1,1
%A A075432 _Reinhard Zumkeller_, Sep 15 2002
%E A075432 More terms (that were already in the b-file) from _Jeppe Stig Nielsen_, Apr 23 2020