A049097 -id:A049097 - cp's OEIS frontend

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

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

A089191 Primes p such that p+1 is cubefree.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 59, 61, 67, 73, 83, 89, 97, 101, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 233, 241, 251, 257, 277, 281, 283, 293, 307, 313, 317, 331, 337, 347, 349, 353, 373, 379, 389, 397, 401, 409

Offset: 1

Cino Hilliard, Dec 08 2003

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+ slightly higher than the p-1 variety.

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 = 2^2*11.
71 is omitted because 71+1 = 2^3*3^2.

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, American Mathematial Monthly 56:1 (1949), pp. 17-19.

Cf. A004709, A049097, A065414, A089189.

filter:= t -> isprime(t) and max(map(s -> s[2], ifactors(t+1)[2]))<3:
select(filter, [2,seq(i,i=3..1000,2)]); # Robert Israel, Mar 18 2018

Select[Prime[Range[100]],Max[Transpose[FactorInteger[#+1]][[2]]]<3&] (* Harvey P. Dale, Jun 06 2013 *)

is(n) = isprime(n) && vecmax(factor(n+1)[,2]) < 3 \\ Amiram Eldar, Feb 16 2021

A120068 Numbers n such that n-th prime + 1 is squarefree.

Original entry on oeis.org

1, 3, 6, 10, 12, 13, 18, 21, 26, 29, 30, 33, 37, 40, 42, 44, 50, 55, 59, 60, 65, 66, 71, 74, 77, 78, 79, 80, 82, 84, 88, 89, 97, 100, 104, 110, 112, 113, 116, 119, 121, 122, 123, 127, 130, 134, 135, 136, 139, 142, 145, 147, 148, 151, 158, 159, 160, 165, 168, 169, 172

Offset: 1

Zak Seidov, Aug 15 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A049097.

Select[Range[200],SquareFreeQ[Prime[#]+1]&] (* Harvey P. Dale, Sep 07 2012 *)

a(n)=primepi(A049097(n))

A153214 Primes p such that p+-2 and p+-3 are not squarefree.

Original entry on oeis.org

47, 1447, 1847, 3701, 6653, 11273, 14947, 15727, 17053, 18493, 21661, 24923, 26647, 29153, 32789, 33023, 38873, 39323, 42437, 42923, 44053, 47527, 47977, 49853, 52027, 52153, 56747, 56873, 59929, 71147, 74189, 79427, 80953, 99277, 99713

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 20 2008

nonn

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

lst={};Do[p=Prime[n];If[ !SquareFreeQ[p-2]&&!SquareFreeQ[p+2]&&!SquareFreeQ[p-3]&&!SquareFreeQ[p+3],AppendTo[lst,p]],{n,3*7!}];lst
Select[Prime[Range[10000]],NoneTrue[#+{-3,-2,2,3},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 27 2019 *)

A268612 Primes p such that p + 2 k, for k = 1..7 are squarefree.

Original entry on oeis.org

29, 83, 101, 191, 227, 389, 443, 479, 587, 641, 659, 677, 983, 1091, 1109, 1289, 1307, 1451, 1487, 2027, 2081, 2153, 2243, 2333, 2351, 2441, 2459, 2477, 2549, 2657, 2729, 2837, 2909, 2927, 2999, 3089, 3251, 3359, 3449, 3557, 3593, 4007

Offset: 1

Zak Seidov, Feb 08 2016

nonn

Eight consecutive odd numbers starting with p are squarefree.
This is the longest set as p+16 in all cases is divisible by 9.
All terms are congruent to 11 mod 18.

Zak Seidov, Table of n, a(n) for n = 1..1592

Cf. A049097, A049233.

[p: p in PrimesUpTo(4500) | forall{k: k in [1..7] | IsSquarefree(p+2*k)}]; // Vincenzo Librandi, Feb 09 2016

Select[Prime[Range[600]],AllTrue[#+2*Range[7],SquareFreeQ]&] (* Harvey P. Dale, Oct 19 2022 *)

A268614 Primes p such that p + 1 and p + 2 are squarefree.

Original entry on oeis.org

5, 13, 29, 37, 41, 101, 109, 113, 137, 157, 181, 193, 229, 257, 281, 317, 353, 389, 397, 401, 409, 433, 461, 509, 541, 569, 613, 617, 641, 653, 661, 677, 757, 761, 769, 797, 821, 829, 857, 877, 937, 941, 977, 1009, 1021, 1093, 1109, 1117, 1129, 1153, 1193

Offset: 1

Zak Seidov, Feb 08 2016

nonn

All terms are == 1 mod 4, hence in all cases p+3 is divisible by 4 (and is not squarefree).

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

Intersection of A049097 and A049233.

Cf. A002144, A005117.

[p: p in PrimesUpTo(1500) | IsSquarefree(p+1) and IsSquarefree(p+2)]; // Vincenzo Librandi, Feb 09 2016

Select[Prime[Range[1000]], SquareFreeQ[# + 1] && SquareFreeQ[# + 2] &]

isok(p) = isprime(p) && issquarefree(p+1) && issquarefree(p+2); \\ Michel Marcus, Apr 01 2021

A340154 Primes p such that p == 5 (mod 6) and p+1 is squarefree.

Original entry on oeis.org

5, 29, 41, 101, 113, 137, 173, 257, 281, 317, 353, 389, 401, 461, 509, 569, 617, 641, 653, 677, 761, 797, 821, 857, 929, 941, 977, 1109, 1181, 1193, 1217, 1229, 1289, 1301, 1361, 1373, 1409, 1433, 1481, 1553, 1613, 1697, 1721, 1877, 1901, 1913, 1973, 2081, 2129

Offset: 1

Amiram Eldar, Dec 29 2020

Clary and Fabrykowski (2004) proved that this sequence is infinite, and that its relative density in the sequence of primes of the form 6*k+5 (A007528) is 4*A/5 = 0.29916465..., where A is Artin's constant (A005596).

			5 is a term since it is prime, 5 == 5 (mod 6), and 5+1 = 6 = 2*3 is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Stuart Clary and Jacek Fabrykowski, Arithmetic progressions, prime numbers, and squarefree integers, Czechoslovak Mathematical Journal, Vol. 54, No. 4 (2004), pp. 915-927.

Intersection of A007528 and A049097.

Cf. A005117, A005596.

Select[Range[2000], Mod[#, 6] == 5 && PrimeQ[#] && SquareFreeQ[# + 1] &]

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

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A089191 Primes p such that p+1 is cubefree.

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A120068 Numbers n such that n-th prime + 1 is squarefree.

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Mathematica

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A153214 Primes p such that p+-2 and p+-3 are not squarefree.

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Mathematica

A268612 Primes p such that p + 2 k, for k = 1..7 are squarefree.

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Magma

Mathematica

A268614 Primes p such that p + 1 and p + 2 are squarefree.

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Magma

Mathematica

PARI

A340154 Primes p such that p == 5 (mod 6) and p+1 is squarefree.

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Mathematica