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A075432
Primes with no squarefree neighbors.
14
17, 19, 53, 89, 97, 127, 149, 151, 163, 197, 199, 233, 241, 251, 269, 271, 293, 307, 337, 349, 379, 449, 487, 491, 521, 523, 557, 577, 593, 631, 701, 727, 739, 751, 773, 809, 811, 881, 883, 919, 953, 991, 1013, 1049, 1051, 1061, 1063, 1097, 1151, 1171, 1249
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
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
Intersection of A000040 and A281192.
Sequence in context: A243437 A144709 A132239 * A232882 A232878 A226681
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