A258801 -id:A258801 - cp's OEIS frontend

A287591 Carmichael numbers k such that k-2 and k+2 are both primes.

656601, 25536531021, 8829751133841, 60561233400921, 79934093254401, 352609909731201, 598438077923841, 976515437206401, 2122162714918401, 2789066007968241, 3767175573114801, 7881891474971361, 10740122274670881, 11512252145095521, 16924806963384321

Offset: 1

Amiram Eldar, May 26 2017

nonn

Rotkiewicz conjectured that there are infinitely many Carmichael numbers k such that k-2 or k+2 are primes.

The terms were calculated using Pinch's tables of Carmichael numbers (see link below).

			656601 is in the sequence since it is a Carmichael number (A002997) and both 656599 and 656603 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..282 (terms below 10^22, calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
R. G. E. Pinch, Tables relating to Carmichael numbers.
Andrzej Rotkiewicz, On pseudoprimes having special forms and a solution of K. Szymiczek's problem, Acta Mathematica Universitatis Ostraviensis, Vol. 13, No. 1 (2005), pp. 57-71.

Cf. A002997, A057942, A272754.

Subsequence of A258801.

A205947 Carmichael numbers not congruent to 1 modulo 6.

Original entry on oeis.org

561, 2465, 62745, 162401, 656601, 1909001, 5444489, 11921001, 19384289, 26719701, 45318561, 84350561, 151530401, 174352641, 221884001, 230996949, 275283401, 434932961, 662086041, 684106401, 689880801, 710382401

Offset: 1

José María Grau Ribas, Feb 02 2012

nonn

These numbers are very sparse; most Carmichael numbers are 1 mod 6. - Charles R Greathouse IV, May 02 2012
Not known to be infinite, see Matomäki. - Charles R Greathouse IV, Jun 13 2012
From Robert Israel, Jul 20 2015: (Start)
Now known to be infinite, see Wright.
No member of this sequence is divisible by any prime of the form 6k+1, hence all prime factors for this sequence are members of A045410. (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Kaisa Matomäki, Carmichael numbers in arithmetic progressions, Journal of the Australian Mathematical Society 94:2 (2013), pp. 268-275.
T. Wright, Infinitely many Carmichael numbers in arithmetic progressions, Bull. London Math. Soc. (2013) 45 (5): 943-952. arXiv:1212.5850

Cf. A002997, A045410, A258801.

korselt:= proc(n) uses numtheory; local p;
  if isprime(n) or not issqrfree(n) then return false fi;
  for p in factorset(n) do
     if n-1 mod (p-1) <> 0 then return false fi
  od;
  true
end proc:
select(korselt, [seq(seq(6*i+j,j=[3,5]),i=1..10^5)]); # Robert Israel, Jul 20 2015

Select[Range[100000], !PrimeQ[#] && IntegerQ[(#-1)/CarmichaelLambda[#]] && !Mod[#,6]==1&]

Korselt(n,f=factor(n))=for(i=1,#f[,1],if(f[i,2]>1||(n-1)%(f[i,1]-1),return(0)));1
list(lim)={
  my(v=List(),p=2);
  forstep(n=561,lim,[12,6],
    if(Korselt(n),listput(v,n))
  );
  forprime(q=3,lim,
    forstep(n=p+if(p%6<5,4,6),q-2,6,
      if(Korselt(n),listput(v,n))
    );
    p=q
  );
  vecsort(Vec(v))
}; \\ Charles R Greathouse IV, Apr 25 2012

Wright shows that there are at least x^(K/(log log log x)^2) terms up to x, for an explicitly computable (though not computed) constant K. - Charles R Greathouse IV, Jul 20 2015

A306451 Non-coprime pseudoprimes or primes to base 3: numbers k that are multiples of 3 and are such that k divides 3^k - 3.

Original entry on oeis.org

3, 6, 66, 561, 726, 7107, 8205, 8646, 62745, 100101, 140097, 166521, 237381, 237945, 566805, 656601, 876129, 1053426, 1095186, 1194285, 1234806, 1590513, 1598871, 1938021, 2381259, 2518041, 3426081, 4125441, 5398401, 5454681, 5489121, 5720331, 5961441

Offset: 1

Jianing Song, Feb 17 2019

nonn

Union of {3} and (A122780 - {1} - A005935).
Numbers of the form 3*m such that 3^(3*m-1) == 1 (mod m).
The squarefree terms are listed in A306450.

Jianing Song, Table of n, a(n) for n = 1..163 (all terms below 10^9)

Cf. A005935, A006935, A122780, A306450.

A258801 is a subsequence.

forstep(n=3, 1e7, 3, if(Mod(3, n)^n==3, print1(n, ", ")))

66 is a term because 66 divides 3^66 - 3 = 3*(3^65 - 1) = 3*(3^5 - 1)*(3^60 + 3^55 + ... + 3^5 + 1) and 66 is divisible by 3.

A216911 Numbers that are a prime factor of a Carmichael number divisible by 3, listed in the order in which they appear in the prime factorizations of such Carmichael numbers.

Original entry on oeis.org

3, 11, 17, 5, 47, 89, 101, 197, 29, 263, 521, 1559, 173, 3011, 71, 641, 1277, 53, 317, 4583, 617, 4019, 401, 3041, 41, 479, 3347, 131, 10427, 4643, 1301, 419, 6689, 5531, 281, 55217, 251, 2417, 4001, 491, 1601, 3137, 449, 3617, 107, 2969, 4211, 6737, 1061

Offset: 1

Marius Coman, Sep 20 2012

nonn

Every term, other than a(1)=3, is a prime of the form 6*k - 1.

			The first few Carmichael numbers that are divisible by 3 (A258801), their prime factorizations, and the terms of this sequence are as follows:
.
  m | A258801(m) & prime factorization | prime factors not appearing earlier
  --+----------------------------------+----------------------------------------
  1 |        561 = 3 * 11 *  17        | a(1)  =    3,  a(2)  = 11,   a(3) = 17;
  2 |      62745 = 3 *  5 *  47 *   89 | a(4)  =    5,  a(5)  = 47,   a(6) = 89;
  3 |     656601 = 3 * 11 * 101 *  197 | a(7)  =  101,  a(8)  = 197; ...

Charles R Greathouse IV, Table of n, a(n) for n = 1..3744
Eric Weisstein's World of Mathematics, Carmichael Number

Cf. A002997 (Carmichael numbers), A258801 (Carmichael numbers divisible by 3).

a(26), a(34), a(43) corrected by Charles R Greathouse IV, Sep 20 2012

A308086 Carmichael numbers c such that c-4, c-2 and c+2 are primes.

Original entry on oeis.org

656601, 11512252145095521, 35151891169379601, 89283676825965441, 209606994019068801, 584047819872236721, 627126355430628801, 1107574117930742001, 1152431453119654401, 2990125943388676401, 6919232969930803761

Offset: 1

Rick L. Shepherd, May 11 2019

nonn

Subsequence of A287591 (Carmichael numbers that are arithmetic means of cousin primes). Calculated from Amiram Eldar's table in that sequence. The Carmichael numbers here are contained within intervals defined by prime triples of the form (p, p+2, p+6); therefore, for each term, four consecutive odd numbers are prime, prime, Carmichael number (divisible by 3), then prime. None of the terms of A287591 available so far are contained within intervals defined by prime triplets of the form (p, p+4, p+6). Is that possible? If so, is it also possible for a Carmichael number to be immediately preceded and succeeded by twin primes, i.e., to be "contained" in a prime quadruplet? (Such Carmichael numbers would necessarily be multiples of 15.)

			656601 = 3*11*101*197 is a term because 656597 and 656599 are twin primes, 656601 is a Carmichael number, and 656603 is also a prime.

Amiram Eldar, Table of n, a(n) for n = 1..36 (terms below 10^22, calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Cf. A002997, A007530, A022004, A022005, A230715, A258801, A287591.

More terms from Amiram Eldar, Jul 02 2019

cp's OEIS Frontend

A287591 Carmichael numbers k such that k-2 and k+2 are both primes.

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A205947 Carmichael numbers not congruent to 1 modulo 6.

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A306451 Non-coprime pseudoprimes or primes to base 3: numbers k that are multiples of 3 and are such that k divides 3^k - 3.

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A216911 Numbers that are a prime factor of a Carmichael number divisible by 3, listed in the order in which they appear in the prime factorizations of such Carmichael numbers.

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A308086 Carmichael numbers c such that c-4, c-2 and c+2 are primes.

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