OFFSET
1,1
COMMENTS
Primes p such that both p-1 and p+1 are divisible by a square greater than 1. - N. J. A. Sloane, Jul 19 2024Complement of A075430 in A000040.
From Ludovicus (luiroto(AT)yahoo.com), Dec 07 2009: (Start)
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:
10^5: 2421 / 9592 = 0.2523978315
10^6: 19812 / 78498 = 0.2523885958
10^7: 167489 / 664579 = 0.2520227091
10^8: 1452678 / 5761455 = 0.2521373507
10^9: 12817966 / 50847534 = 0.2520862860
10^10: 114713084 / 455052511 = 0.2520875750
10^11: 1038117249 / 4118054813 = 0.2520892256
It comes close to the expected 1-2A. (End)
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
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Pieter Moree, Artin's primitive root conjecture -a survey -, arXiv:math/0412262 [math.NT], 2004-2012.
Carlos Rivera, Conjecture 65. Non-Euclidean primes, The Prime Puzzles and Problems Connection.
FORMULA
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
EXAMPLE
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
MAPLE
filter:= n -> isprime(n) and not numtheory:-issqrfree(n+1) and not numtheory:-issqrfree(n-1):
select(filter, [seq(2*i+1, i=1..1000)]); # Robert Israel, Aug 27 2014
MATHEMATICA
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 *)
Select[Prime[Range[300]], !SquareFreeQ[#-1]&&!SquareFreeQ[#+1]&] (* Harvey P. Dale, Apr 24 2014 *)
PROG
(Haskell)
a075432 n = a075432_list !! (n-1)
a075432_list = f [2, 4 ..] where
f (u:vs@(v:ws)) | a008966 v == 1 = f ws
| a008966 u == 1 = f vs
| a010051' (u + 1) == 0 = f vs
| otherwise = (u + 1) : f vs
-- Reinhard Zumkeller, May 04 2013
(PARI) is(n)=!issquarefree(if(n%4==1, n+1, n-1)) && isprime(n) \\ Charles R Greathouse IV, Aug 27 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Sep 15 2002
EXTENSIONS
More terms (that were already in the b-file) from Jeppe Stig Nielsen, Apr 23 2020
STATUS
approved