A039787 -id:A039787 - cp's OEIS frontend

A049091 Duplicate of A039787.

2, 3, 7, 11, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 131, 139, 167, 179, 191, 211

Offset: 1

dead

A049097 Primes p such that p+1 is squarefree.

2, 5, 13, 29, 37, 41, 61, 73, 101, 109, 113, 137, 157, 173, 181, 193, 229, 257, 277, 281, 313, 317, 353, 373, 389, 397, 401, 409, 421, 433, 457, 461, 509, 541, 569, 601, 613, 617, 641, 653, 661, 673, 677, 709, 733, 757, 761, 769, 797, 821, 829, 853, 857

Offset: 1

Labos Elemer

nonn

Numbers k such that core(sigma(k)) = k + 1 where core(k) is the squarefree part of k (A007913). - Benoit Cloitre, May 01 2002

This sequence is infinite and its relative density in the sequence of primes is equal to Artin's constant (A005596): Product_{p prime} (1-1/(p*(p-1))) = 0.373955... (Mirsky, 1949). - Amiram Eldar, Dec 29 2020

			29 is included since 29 + 1 = 30 = 2*3*5 is squarefree.
17 is not here because 18 is divisible by a square, 9.

Robert Israel, Table of n, a(n) for n = 1..10000
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. A000040, A005117, A039787.

Cf. A000203, A005596, A007913, A077067, A160696.

[ p: p in PrimesUpTo(900) | IsSquarefree(p+1) ]; // Vincenzo Librandi, Dec 25 2010

N:= 10000; # to get all entries up to N
A049097:= select(t -> isprime(t) and numtheory:-issqrfree(t+1), [2, seq(1+2*k,k=1..floor((N-1)/2))]); # Robert Israel, May 11 2014

Select[Prime[Range[100]], SquareFreeQ[# + 1] &] (* Zak Seidov, Feb 08 2016 *)

lista(nn) = forprime(p=1, nn, if (issquarefree(p+1), print1(p, ", "))); \\ Michel Marcus, Jan 08 2015

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

a(n) = A077067(n)-1. - Zak Seidov, Mar 19 2016

A075432 Primes with no squarefree neighbors.

Original entry on oeis.org

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

A089189 Primes p such that p-1 is cubefree.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 127, 131, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 239, 263, 269, 277, 283, 293, 307, 311, 317, 331, 347, 349, 359, 367, 373, 383

Offset: 1

Cino Hilliard, Dec 08 2003 and Reinhard Zumkeller, Aug 11 2004

The ratio of the count of primes p <= n such that p-1 is cubefree to the count of primes <= n converges to 0.69.. . This implies that roughly 70% of the primes less one are cubefree. This compares to about 0.37 of the primes less one are squarefree.

More accurately, the density of this sequence within the primes is Product_{p prime} (1-1/(p^2*(p-1))) = 0.697501... (A065414) (Mirsky, 1949). - Amiram Eldar, Feb 16 2021

			43 is included because 43-1 = 2*3*7.
41 is omitted because 41-1 = 2^3*5.
97 is omitted because 96 = 2^5*3 since higher powers are also tested for exclusion.

Reinhard Zumkeller and Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, American Mathematial Monthly 56:1 (1949), pp. 17-19.

Cf. A004709, A039787, A065414, A097380, A089194 (subsequence).

a097375 n = a097375_list !! (n-1)
a097375_list = filter ((== 1) . a212793 . (subtract 1)) a000040_list
-- Reinhard Zumkeller, May 27 2012

filter:= p -> isprime(p) and max(seq(t[2],t=ifactors(p-1)[2]))<=2:
select(filter, [2,seq(2*i+1,i=1..1000)]); # Robert Israel, Sep 11 2014

f[n_]:=Module[{a=m=0},Do[If[FactorInteger[n][[m,2]]>2,a=1],{m,Length[FactorInteger[n]]}];a]; lst={};Do[p=Prime[n];If[f[p-1]==0,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 15 2009 *)
Select[Prime[Range[100]],Max[Transpose[FactorInteger[#-1]][[2]]]<3&] (* Harvey P. Dale, Feb 05 2012 *)

lista(nn) = forprime(p=2, nn, f = factor(p-1)[,2]; if ((#f == 0) || vecmax(f) < 3, print1(p, ", "));) \\ Michel Marcus, Sep 11 2014

A212793(a(n) - 1) = 1. - Reinhard Zumkeller, May 27 2012

Corrected and extended by Harvey P. Dale, Feb 05 2012

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

A049092 Primes p such that p-1 is not squarefree.

Original entry on oeis.org

5, 13, 17, 19, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 127, 137, 149, 151, 157, 163, 173, 181, 193, 197, 199, 229, 233, 241, 251, 257, 269, 271, 277, 281, 293, 307, 313, 317, 337, 349, 353, 373, 379, 389, 397, 401, 409, 421, 433, 449, 457, 461, 487

Offset: 1

Labos Elemer

nonn

Primes p with mu(p-1)=0, where mu is the Möbius function. - T. D. Noe, Nov 03 2003
Primes p such that the sum of the primitive roots of p (see A088144) is 0 mod p. - Jon Wharf, Mar 12 2015
The relative density of this sequence within the primes is 1 - A005596 = 0.626044... - Amiram Eldar, Feb 10 2021

			p = 257 is here because p-1 = 256 = 2^8.
p = 997 is here because p-1 = 996 = 3*(2^2)*83.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Moebius Function.

Cf. A005596, A039787, A078330 (primes p with mu(p-1)=-1), A088179 (primes p such that mu(p-1)=1), A089451 (mu(p-1) for prime p), A145199.

[ p: p in PrimesUpTo(500) | not IsSquarefree(p-1) ]; // Vincenzo Librandi, Mar 12 2015

Select[Prime[Range[400]], MoebiusMu[ #-1]==0&]

forprime(p=2,500,if(!issquarefree(p-1),print(p))) \\ Michael B. Porter, Mar 16 2015

a(n) = A145199(n) + 1. - Amiram Eldar, Feb 10 2021

A049149 Numbers k such that the Euler totient function phi(k) is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 14, 18, 22, 23, 31, 43, 46, 47, 49, 59, 62, 67, 71, 79, 83, 86, 94, 98, 103, 107, 118, 121, 131, 134, 139, 142, 158, 166, 167, 179, 191, 206, 211, 214, 223, 227, 239, 242, 262, 263, 278, 283, 311, 331, 334, 347, 358, 359, 367, 382, 383

Offset: 1

Labos Elemer

nonn

Consists of 1, 2, 4, p, p^2, 2p, and 2p^2, where p are the odd primes from A039787. - Ivan Neretin, Aug 24 2016

			a(17) = 49 is here because phi(49) = 42 = 2*3*7 is squarefree. Primes p, such that p-1 is squarefree are included.

Ivan Neretin, Table of n, a(n) for n = 1..10000
William D. Banks and Francesco Pappalardi, Values of the Euler function free of kth powers, Journal of Number Theory, Vol. 120, No. 2 (2006), pp. 326-348.
Francesco Pappalardi, Filip Saidak and Igor E. Shparlinski, Square-free values of the Carmichael function, Journal of Number Theory, Vol. 103, No. 1 (2003), pp. 122-131.

Cf. A000010, A000720, A005117, A005596, A039787, A013929.

Select[Range[100], MoebiusMu[EulerPhi[#]] != 0 &]

isok(n) = issquarefree(eulerphi(n)); \\ Michel Marcus, Aug 24 2016

The number of terms not exceeding k is (3*a/2) * pi(k) + O(k/(log(k)^c)), where pi(k) = A000720(k), c is any constant > 0, and a = 0.373955... is Artin's constant (A005596) (Pappalardi et al., 2003; Banks and Pappalardi, 2006). - Amiram Eldar, Jul 28 2020

Corrected by T. D. Noe, Oct 25 2006

A089200 Primes p such that p-1 is divisible by a cube.

Original entry on oeis.org

17, 41, 73, 89, 97, 109, 113, 137, 163, 193, 233, 241, 251, 257, 271, 281, 313, 337, 353, 379, 401, 409, 433, 449, 457, 487, 521, 541, 569, 577, 593, 601, 617, 641, 673, 751, 757, 761, 769, 809, 811, 857, 881, 919, 929, 937, 953, 977

Offset: 1

Cino Hilliard, Dec 08 2003

This sequence is infinite and its relative density in the sequence of primes is 1 - Product_{p prime} (1-1/(p^2*(p-1))) = 1 - A065414 = 0.30249864150363409671... (Jakimczuk, 2024). - Amiram Eldar, Jul 20 2024

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Numbers of the form p-1 where p is prime, ResearchGate preprint, 2024.

Cf. A039787, A046099, A065414.

f[n_]:=Max[Last/@FactorInteger[n]]; lst={};Do[p=Prime[n];If[f[p-1]>=3,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)
Select[Prime[Range[200]],Count[Transpose[FactorInteger[#-1]][[2]], ?(#>2&)]>0&] (* _Harvey P. Dale, Jan 01 2012 *)

ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) }
powerfreep3(n,p,k) = { c=0; pc=0; forprime(x=2,n, pc++; if(ispowerfree(x+k,p)==0, c++; print1(x","); ) ); print(); print(c","pc","c/pc+.0) }

A097379 Numbers m such that 1+SquareFreeKernel(m) is prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 30, 32, 36, 40, 42, 44, 46, 48, 50, 54, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 90, 92, 96, 100, 102, 106, 108, 116, 120, 126, 128, 130, 132, 138, 140, 144, 150, 156, 160, 162, 164, 166, 168, 176, 178, 180, 184, 190, 192

Offset: 1

Reinhard Zumkeller, Aug 11 2004

nonn

			m = 100 = (2*5)^2 -> A076618(100) = 1+2*5 = 11 = A000040(5), therefore 100 is a term.

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

Cf. A007947, A076618, A039787, A097380, A097381, A359213.

Select[Range[200], PrimeQ[1 + Times @@ FactorInteger[#][[;; , 1]]] &] (* Amiram Eldar, Feb 01 2024 *)

is(n) = isprime(1 + vecprod(factor(n)[, 1])); \\ Amiram Eldar, Feb 01 2024

A076618(a(n)) = A007947(a(n))+1 is prime.

A049231 Primes p such that p - 2 is squarefree.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 23, 31, 37, 41, 43, 53, 59, 61, 67, 71, 73, 79, 89, 97, 103, 107, 109, 113, 131, 139, 151, 157, 163, 167, 179, 181, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 283, 293, 307, 311, 313, 331, 337, 347, 349, 359, 367, 373

Offset: 1

Labos Elemer

nonn

This sequence is infinite and its relative density in the sequence of the primes is equal to 2 * Product_{p prime} (1-1/(p*(p-1))) = 2 * A005596 = 0.747911... (Mirsky, 1949). - Amiram Eldar, Feb 27 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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. A005117, A005596, A039787, A049233.

Select[Prime[Range[100]],SquareFreeQ[#-2]&] (* Harvey P. Dale, Mar 03 2018 *)

isok(p) = isprime(p) && issquarefree(p-2); \\ Michel Marcus, Dec 31 2013

Primes p such that abs(mu(p-2)) = 1.

Definition corrected by Michel Marcus, Dec 31 2013

cp's OEIS Frontend

A049091 Duplicate of A039787.

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A049097 Primes p such that p+1 is squarefree.

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A075432 Primes with no squarefree neighbors.

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A089189 Primes p such that p-1 is cubefree.

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

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A049092 Primes p such that p-1 is not squarefree.

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A049149 Numbers k such that the Euler totient function phi(k) is squarefree.

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