A075430 -id:A075430 - cp's OEIS frontend

A075432 Primes with no squarefree neighbors.

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

Reinhard Zumkeller, Sep 15 2002

nonn

Complement 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
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

			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

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.

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

Intersection of A000040 and A281192.

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

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

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 *)

is(n)=!issquarefree(if(n%4==1, n+1, n-1)) && isprime(n) \\ Charles R Greathouse IV, Aug 27 2014

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

More terms (that were already in the b-file) from Jeppe Stig Nielsen, Apr 23 2020

A240475 Primes that are midway between the closest flanking squarefree numbers.

Original entry on oeis.org

2, 17, 19, 53, 89, 163, 197, 199, 233, 251, 269, 271, 293, 307, 337, 379, 449, 487, 491, 521, 557, 593, 631, 701, 739, 751, 809, 811, 881, 883, 919, 953, 991, 1013, 1049, 1061, 1063, 1097, 1151, 1171, 1279, 1459, 1471, 1493, 1531, 1549, 1567, 1601, 1637

Offset: 1

Chris Boyd, Apr 06 2014

nonn

Primes for which the corresponding A240473(m) is equal to A240474(m).
Primes equal to the average of the closest flanking squarefree numbers.
Primes equal to the average of three consecutive squarefree numbers.
Most terms are such that a(n)+2 and a(n)-2 are the closest squarefree numbers. The first term > 2 for which this is not the case is a(880) = 47527.
494501773, 765921647, 930996623 are the terms < 10^9 that also belong to A176141.

			19 is a term because it is midway between the closest flanking squarefree numbers 17 and 21.
On the other hand, 29 is not a term because it is not midway between the closest flanking squarefree numbers 26 and 30.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A000040, A075430, A075432, A166003, A176141, A240473, A240474, A240476.

Select[Mean/@Partition[Select[Range[2000],SquareFreeQ],3,1],PrimeQ] (* Harvey P. Dale, Jul 27 2024 *)

forprime(p=1,1650,forstep(j=p-1,1,-1,if(issquarefree(j),L=j;break));for(j=p+1,2*p,if(issquarefree(j),G=j;break));if(G-p==p-L,print1(p", ")))

A240476 Primes that are not midway between the closest flanking squarefree numbers.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 193, 211, 223, 227, 229, 239, 241, 257, 263, 277, 281, 283, 311, 313, 317, 331, 347, 349, 353

Offset: 1

Chris Boyd, Apr 06 2014

nonn

Primes for which the corresponding A240473(m) is not equal to A240474(m).
Primes not equal to the average of the closest flanking squarefree numbers.
Primes not equal to the average of three consecutive squarefree numbers.
Complement of A240475 relative to A000040.

			29 is a term because it is not midway between the closest flanking squarefree numbers 26 and 30.
On the other hand, 19 is not a term because it is midway between the closest flanking squarefree numbers 17 and 21.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A000040, A075430, A075432, A240473, A240474, A240475

forprime(p=1,353,forstep(j=p-1,1,-1,if(issquarefree(j),L=j;break));for(j=p+1,2*p,if(issquarefree(j),G=j;break));if(G-p!=p-L,print1(p", ")))

A075431 Primes of the form n+mu(n), where mu is the Moebius function (A008683).

Original entry on oeis.org

2, 7, 11, 23, 29, 41, 47, 59, 83, 101, 107, 109, 113, 137, 167, 173, 179, 181, 211, 227, 229, 257, 263, 281, 317, 331, 347, 353, 359, 373, 383, 401, 409, 433, 463, 467, 479, 503, 547, 563, 571, 587, 601, 617, 641, 653, 677, 691, 709, 719, 761, 821, 829, 839, 853

Offset: 1

Reinhard Zumkeller, Sep 15 2002

nonn

Subsequence of A075430.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A063015, A039787, A049097.

isok(p)={isprime(p) && (moebius(p+1) == -1 || moebius(p-1) == 1)} \\ Andrew Howroyd, Apr 20 2021

Terms a(41) and beyond from Andrew Howroyd, Apr 20 2021

cp's OEIS Frontend

A075432 Primes with no squarefree neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A240475 Primes that are midway between the closest flanking squarefree numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A240476 Primes that are not midway between the closest flanking squarefree numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A075431 Primes of the form n+mu(n), where mu is the Moebius function (A008683).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions