A049097 -id:A049097 - 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

A153213 Primes p such that both p-2 and p+2 are not squarefree.

Original entry on oeis.org

2, 47, 173, 277, 727, 839, 929, 1181, 1423, 1447, 1523, 1627, 1811, 1847, 1861, 1973, 2207, 2297, 2423, 2693, 3323, 3701, 3719, 3877, 4327, 4363, 4457, 4673, 4691, 4903, 5227, 5573, 5821, 5927, 6173, 6221, 6323, 6473, 6577, 6653, 7027, 7103, 7477, 7823

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 20 2008

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A049282, A049097, A039787, A075432, A153207, A153208, A153209, A153210.

lst={};Do[p=Prime[n];If[ !SquareFreeQ[p-2]&&!SquareFreeQ[p+2],AppendTo[lst,p]],{n,7!}];lst

is(n)=isprime(n) && !issquarefree(n-2) && !issquarefree(n+2) \\ Charles R Greathouse IV, Nov 05 2017

A049098 Primes p such that p+1 is divisible by a square.

Original entry on oeis.org

3, 7, 11, 17, 19, 23, 31, 43, 47, 53, 59, 67, 71, 79, 83, 89, 97, 103, 107, 127, 131, 139, 149, 151, 163, 167, 179, 191, 197, 199, 211, 223, 227, 233, 239, 241, 251, 263, 269, 271, 283, 293, 307, 311, 331, 337, 347, 349, 359, 367, 379, 383, 419, 431, 439, 443

Offset: 1

Labos Elemer

Numbers m such that A010051(m)*(1-A008966(m+1)) = 1. - Reinhard Zumkeller, May 21 2009

This sequence is infinite and its relative density in the sequence of primes is equal to 1 - Product_{p prime} (1-1/(p*(p-1))) = 1 - A005596 = 0.626044... (Mirsky, 1949). - Amiram Eldar, Feb 14 2021

			31 is a term because 32 is divisible by a square, 16.
101 is not a term because 102 = 2*3*17 is squarefree.

T. D. Noe, Table of n, a(n) for n = 1..1000
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 17-19.

Cf. A005596, A008966, A010051, A049097 (complement with respect to A000040), A160696.

a049098 n = a049098_list !! (n-1)
a049098_list = filter ((== 0) . a008966 . (+ 1)) a000040_list
-- Reinhard Zumkeller, Oct 18 2011

with(numtheory): a := proc (n) if isprime(n) = true and issqrfree(n+1) = false then n else end if end proc: seq(a(n), n = 1 .. 500); # Emeric Deutsch, Jun 21 2009

Select[Prime[Range[200]],!SquareFreeQ[#+1]&]   (* Harvey P. Dale, Mar 27 2011 *)
Select[Prime[Range[200]], MoebiusMu[# + 1] == 0 &] (* Alonso del Arte, Oct 18 2011 *)

forprime(p=2,1e4,if(!issquarefree(p+1),print1(p", "))) \\ Charles R Greathouse IV, Oct 18 2011

A160696(a(n)) > 1. - Reinhard Zumkeller, May 24 2009

A077067 Squarefree numbers of the form prime + 1.

Original entry on oeis.org

3, 6, 14, 30, 38, 42, 62, 74, 102, 110, 114, 138, 158, 174, 182, 194, 230, 258, 278, 282, 314, 318, 354, 374, 390, 398, 402, 410, 422, 434, 458, 462, 510, 542, 570, 602, 614, 618, 642, 654, 662, 674, 678, 710, 734, 758, 762, 770, 798, 822, 830, 854, 858, 878

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

			A005117(28) = 42 = 2*3*7 is a term as 42 = A000040(13) + 1 = 41+1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A005117, A008864, A049097, A077064, A077066.

Select[Prime[Range[200]]+1,SquareFreeQ] (* Harvey P. Dale, Aug 20 2017 *)

isok(n) = issquarefree(n) && isprime(n-1); \\ Michel Marcus, Mar 22 2016

lista(nn) = forprime(p=2, nn, if (issquarefree(p+1), print1(p+1, ", "))); \\ Michel Marcus, Mar 22 2016

A077066(a(n)) = a(n).

a(n) = A049097(n)+1. - Zak Seidov, Aug 15 2006

A065301 Numbers k such that both k and the sum of its divisors are squarefree numbers.

Original entry on oeis.org

1, 2, 5, 13, 26, 29, 37, 41, 61, 73, 74, 101, 109, 113, 122, 137, 146, 157, 173, 181, 193, 218, 229, 257, 277, 281, 313, 314, 317, 353, 362, 373, 386, 389, 397, 401, 409, 421, 433, 457, 458, 461, 509, 541, 554, 569, 601, 613, 617, 626, 641, 653, 661, 673, 677

Offset: 1

Labos Elemer, Oct 29 2001

nonn

From Amiram Eldar, Mar 08 2025: (Start)
Number k such that A280710(k) * A280710(A000203(k)) = 1, or equivalently, A280710(k) * A280710(A048250(k)) = 1.
Squarefree numbers k whose prime factors are terms of A049097, and the elements of the set {p+1 , p|k} are pairwise coprime. (End)

			For k = 13, sigma(13) = 14 = 2*7 is squarefree.
For k = 26, sigma(26) = 1 + 2 + 13 + 26 = 42 = 2*3*7 is squarefree.
For k = 277 (prime), sigma(277) = 278 = 2*139 is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000203, A008683, A048250, A065299, A065300.

Cf. A049097, A065303, A087248, A087249, A280710.

Select[Range[1000],AllTrue[{#,DivisorSigma[1,#]},SquareFreeQ]&] (* Harvey P. Dale, Aug 09 2014 *)

is(m) = abs(moebius(m))==1 && abs(moebius(sigma(m)))==1 \\ Harry J. Smith, Oct 15 2009

from sympy import divisor_sigma
from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n)==n
print([n for n in range(1, 1001) if issquarefree(n) and issquarefree(divisor_sigma(n,1))]) # Indranil Ghosh, Mar 19 2017

A075430 Primes with a squarefree neighbor.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 101, 103, 107, 109, 113, 131, 137, 139, 157, 167, 173, 179, 181, 191, 193, 211, 223, 227, 229, 239, 257, 263, 277, 281, 283, 311, 313, 317, 331, 347, 353, 359, 367, 373, 383, 389, 397

Offset: 1

Reinhard Zumkeller, Sep 15 2002

nonn

			a(3) = 5 because 6 (the number above) is squarefree.
a(13) = 47 because 46 (the number below) is squarefree.
53 is not in the sequence because both 52 and 54 have squares among their divisors.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)

Union of A039787 and A049097.
Complement of A075432 in A000040.
Cf. A005117.

Select[Prime[Range[100]], Or @@ SquareFreeQ /@ (# + {-1, 1}) &] (* Amiram Eldar, May 07 2025 *)

isok(p) = isprime(p) && (issquarefree(p-1) || issquarefree(p+1)); \\ Michel Marcus, Feb 20 2023

def is_A075430(n): return is_prime(n) and (is_squarefree(n-1) or is_squarefree(n+1)) # D. S. McNeil, Jan 16 2011

A081083 Numbers n such that rad(n+1)=rad(n)+1, where rad(m)=A007947(m) is the squarefree kernel of m.

Original entry on oeis.org

1, 2, 5, 6, 8, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 46, 48, 57, 58, 61, 65, 66, 69, 70, 73, 77, 78, 82, 85, 86, 93, 94, 101, 102, 105, 106, 109, 110, 113, 114, 118, 122, 129, 130, 133, 137, 138, 141, 142, 145, 154, 157, 158, 165, 166, 173, 177, 178, 181

Offset: 1

Reinhard Zumkeller, Mar 04 2003

nonn

Nearly all terms seem to be squarefree, see A081084.

			m=46=2*23=rad(46) and rad(47)=47=46+1=rad(46)+1, therefore 46 is a term;
m=48=3*2^4, rad(48)=6 and rad(49)=rad(7*7)=7=6+1=rad(48)+1, therefore 48 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007947, A049097.

Union of A007674 and A081084.

rad[n_] := Times @@ (First/@ FactorInteger[n]); s = {}; r1= 1; Do[r2 = rad[n]; If[r2 == r1 +1, AppendTo[s, n-1]]; r1 = r2, {n,2, 182}]; s (* Amiram Eldar, Aug 22 2019 *)

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
is(n)=rad(n+1)==rad(n)+1 \\ Charles R Greathouse IV, Aug 08 2013

A160696 Largest k such that k^2 divides prime(n)+1.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 3, 2, 2, 1, 4, 1, 1, 2, 4, 3, 2, 1, 2, 6, 1, 4, 2, 3, 7, 1, 2, 6, 1, 1, 8, 2, 1, 2, 5, 2, 1, 2, 2, 1, 6, 1, 8, 1, 3, 10, 2, 4, 2, 1, 3, 4, 11, 6, 1, 2, 3, 4, 1, 1, 2, 7, 2, 2, 1, 1, 2, 13, 2, 5, 1, 6, 4, 1, 2, 8, 1, 1, 1, 1, 2, 1, 12, 1, 2, 2, 15, 1, 1, 4, 6, 4, 2, 2, 10, 6, 1, 3, 2, 1, 2, 3, 2

Offset: 1

Reinhard Zumkeller, May 24 2009

nonn

A160697 and A160698 give record values and where they occur.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000188, A008864, A049097, A049098.

a(n) = core(prime(n)+1, 1)[2]; \\ Michel Marcus, Nov 06 2022

a(A049097(n)) = 1; a(A049098(n)) > 1;

a(n) = A000188(A008864(n)).

A153215 Primes p such that none of p-2, p-1, p+1, and p+2 is squarefree.

Original entry on oeis.org

727, 1423, 1861, 3719, 6173, 9749, 11321, 13183, 19073, 20873, 23227, 23473, 23827, 26981, 27883, 34351, 35323, 41263, 42677, 44449, 45127, 45523, 47527, 48751, 49727, 52391, 53623, 53849, 68749, 71993, 72559, 78823, 83609, 89227, 92779

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 20 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049282, A049097, A039787, A075432, A153207, A153208, A153209, A153210, A153213, A153214.

<< NumberTheory`NumberTheoryFunctions` lst={}; Do[p=Prime[n];If[ !SquareFreeQ[p-1]&&!SquareFreeQ[p+1]&&!SquareFreeQ[p-2]&&!SquareFreeQ[p+2],AppendTo[lst,p]],{n,4*7!}]; lst

A023510 Greatest exponent in prime-power factorization of prime(n) + 1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 2, 2, 3, 1, 5, 1, 1, 2, 4, 3, 2, 1, 2, 3, 1, 4, 2, 2, 2, 1, 3, 3, 1, 1, 7, 2, 1, 2, 2, 3, 1, 2, 3, 1, 2, 1, 6, 1, 2, 3, 2, 5, 2, 1, 2, 4, 2, 2, 1, 3, 3, 4, 1, 1, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 1, 3, 4, 1, 2, 7, 1, 1, 1, 1, 2, 1, 4, 1, 3, 2, 2, 1, 1, 4, 2, 5, 3, 2, 3, 3, 1, 2, 2

Offset: 1

Clark Kimberling

nonn

If a(n)=1 then prime(a(n)) is a term in A049097. - Zak Seidov, Jul 20 2016

			For n=5, the fifth prime is 11, and the prime factorization of 11 + 1 = 12 is 2^2*3. This has exponents 2 and 1, so a(5) is the largest of these exponents, 2. - _Michael B. Porter_, Jul 20 2016

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A008864, A023509, A049097, A051903.

Table[Max[#[[2]] & /@ FactorInteger[Prime[k] + 1]], {k, 10000}] (* Zak Seidov, Jul 19 2016 *)

a(n) = vecmax(factor(prime(n)+1)[,2]) \\ Michel Marcus, Jul 20 2016

a(n) = A051903(A008864(n)). - Michel Marcus, Jul 20 2016

cp's OEIS Frontend

A075432 Primes with no squarefree neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A153213 Primes p such that both p-2 and p+2 are not squarefree.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A049098 Primes p such that p+1 is divisible by a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A077067 Squarefree numbers of the form prime + 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A065301 Numbers k such that both k and the sum of its divisors are squarefree numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A075430 Primes with a squarefree neighbor.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

A081083 Numbers n such that rad(n+1)=rad(n)+1, where rad(m)=A007947(m) is the squarefree kernel of m.

Views

Author

Keywords

Comments

Examples