A006931 -id:A006931 - cp's OEIS frontend

A135717 a(n) = number of prime divisors of Carmichael numbers A002997(n).

3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 4, 3, 4, 3, 4, 4, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 4, 4, 4, 4, 4, 5, 4, 4, 4, 3, 4, 5, 4, 3, 3, 3, 4, 3, 4, 3, 3, 4, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 3, 4, 3, 3, 4, 4, 4, 4, 4, 3, 3, 4, 3, 3, 4, 3, 4, 4, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 5

Offset: 1

Artur Jasinski, Nov 25 2007

nonn

Number of prime divisors is always >= 3. For the least Carmichael number with n prime factors see A006931.

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

Cf. A002997, A006931.

a(n) = A001221(A002997(n)). - M. F. Hasler, Apr 14 2015

A174612 Number of four-prime Carmichael numbers less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 19, 55, 144, 314, 619, 1179, 2102, 3639, 6042, 9938, 16202, 25758, 40685, 63343, 98253, 151566, 232742

Offset: 0

Michel Lagneau, Mar 23 2010

			For n=5, the smallest Carmichael number with 4 prime factors is 41041 = 7*11*13*41.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 220.

Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22.
Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22. [Local copy, with permission]
R. G. E. Pinch, The Carmichael numbers up to 10^21, Proceedings of Conference on Algorithmic Number Theory 2007.

For k-prime Carmichael numbers up to 10^n for k = 3,4,...,11, see A132195, A174612, A174613, A174614, A174615, A174616, A174617, A299710, A299711.

Cf. A002997, A006931, A055553.

a(0) inserted and a(22) from Claude Goutier added by Amiram Eldar, Apr 19 2024

A174614 Number of six-prime Carmichael numbers less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 99, 459, 1714, 5270, 14401, 36907, 86696, 194306, 414660, 849564, 1681744, 3230120, 6034046

Offset: 0

Michel Lagneau, Mar 23 2010

			For n=9: the smallest Carmichael number with 6 prime factors is 321197185 = 5*19*23*29*37*137.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 220.

Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22.
Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22. [Local copy, with permission]
R. G. E. Pinch, The Carmichael numbers up to 10^21, Proceedings of Conference on Algorithmic Number Theory 2007.

For k-prime Carmichael numbers up to 10^n for k = 3,4,...,11, see A132195, A174612, A174613, A174614, A174615, A174616, A174617, A299710, A299711.

Cf. A002997, A006931, A055553.

a(22) from Claude Goutier added by Amiram Eldar, Apr 19 2024

A174615 Number of seven-prime Carmichael numbers less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 41, 262, 1340, 5359, 19210, 60150, 172234, 460553, 1159167, 2774702, 6363475, 14056367

Offset: 0

Michel Lagneau, Mar 23 2010

			The smallest Carmichael number with 7 prime factors is 5394826801 = 7*13*17*23*31*67*73, and there is one other 10-digit example, so a(10)=2.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 220.

Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22.
Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22. [Local copy, with permission]
R. G. E. Pinch, The Carmichael numbers up to 10^21, Proceedings of Conference on Algorithmic Number Theory 2007.

For k-prime Carmichael numbers up to 10^n for k = 3,4,...,11, see A132195, A174612, A174613, A174614, A174615, A174616, A174617, A299710, A299711.

Cf. A002997, A006931, A055553.

a(22) from Claude Goutier added by Amiram Eldar, Apr 19 2024

A174616 Number of eight-prime Carmichael numbers less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 89, 655, 3622, 16348, 63635, 223997, 720406, 2148017, 6015901, 16005646

Offset: 0

Michel Lagneau, Mar 23 2010

			The smallest Carmichael number with 8 prime factors is 232250619601 = 7*11*13*17*31*37*73*163, and there are 6 others, so a(12) = 7.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 220.

Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22.
Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22. [Local copy, with permission]
R. G. E. Pinch, The Carmichael numbers up to 10^21, Proceedings of Conference on Algorithmic Number Theory 2007.

For k-prime Carmichael numbers up to 10^n for k = 3,4,...,11, see A132195, A174612, A174613, A174614, A174615, A174616, A174617, A299710, A299711.

Cf. A002997, A006931, A055553.

a(22) from Claude Goutier added by Amiram Eldar, Apr 19 2024

A174617 Number of nine-prime Carmichael numbers less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 27, 170, 1436, 8835, 44993, 196391, 762963, 2714473, 8939435

Offset: 0

Michel Lagneau, Mar 23 2010

			The smallest Carmichael number with 9 prime factors is 9746347772161 = 7*11*13*17*19*31*37*41*641, so a(13)=1..

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 220.

Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22.
Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22. [Local copy, with permission]
R. G. E. Pinch, The Carmichael numbers up to 10^21, Proceedings of Conference on Algorithmic Number Theory 2007.
R. G. E. Pinch, The Carmichael numbers up to 10^21, Poster, Proceedings of Conference on Algorithmic Number Theory 2007.

For k-prime Carmichael numbers up to 10^n for k = 3,4,...,11, see A132195, A174612, A174613, A174614, A174615, A174616, A174617, A299710, A299711.

Cf. A002997, A006931, A055553.

a(22) from Claude Goutier added by Amiram Eldar, Apr 19 2024

A299711 Number of eleven-prime Carmichael numbers less than 10^n.

Original entry on oeis.org

1, 49, 576, 5804, 42764, 262818

Offset: 17

Tim Johannes Ohrtmann, Feb 17 2018

			60977817398996785 = 5*7*17*19*23*37*53*73*79*89*233 is the only Carmichael number with eleven prime factors below 10^17, so a(17) = 1.

Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22.
Claude Goutier, Text file readme.text summarizing enumeration of Carmichael numbers up to 10^22. [Local copy, with permission]
Richard Pinch, Tables relating to Carmichael numbers.

For k-prime Carmichael numbers up to 10^n for k = 3,4,...,11, see A132195, A174612, A174613, A174614, A174615, A174616, A174617, A299710, A299711.

Cf. A002997, A006931, A055553.

a(22) from Claude Goutier added by Amiram Eldar, Apr 19 2024

A112431 Carmichael numbers equal to the product of 8 primes.

Original entry on oeis.org

232250619601, 306177962545, 432207073585, 576480525985, 658567396081, 689702851201, 747941832001, 1013666981041, 1110495895201, 1111586883121, 1286317859905, 1292652236161, 1341323384401, 1471186523521, 1567214060545

Offset: 1

Shyam Sunder Gupta, Dec 11 2005

			a(1) = 232250619601 = 7*11*13*17*31*37*73*163.

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

Cf. A002997, A087788, A141711, A074379, A112428, A112429, A112430, A112432, A006931.

is(n)={bigomega(n)==8&&is_A002997(n)} \\ M. F. Hasler, Apr 14 2015

A141711 Carmichael numbers with more than 3 prime factors.

Original entry on oeis.org

41041, 62745, 63973, 75361, 101101, 126217, 172081, 188461, 278545, 340561, 449065, 552721, 656601, 658801, 670033, 748657, 825265, 838201, 852841, 997633, 1033669, 1050985, 1082809, 1569457, 1773289, 2100901, 2113921, 2433601

Offset: 1

M. F. Hasler, Jul 01 2008

nonn

Sequence A087788 gives Carmichael numbers with exactly 3 prime factors; since they cannot have fewer (cf. references in A002997), this sequence is the complement of A087788 in A002997.

The terms preceding a(17) = 825265 = A006931(5) have exactly 4 prime factors. See A112428 - A112432 for Carmichael numbers with exactly 5, ..., 9 prime factors. - M. F. Hasler, Apr 14 2015

			a(17)=825265 is the least Carmichael number having more than 4 divisors, thus the sequence differs from A074379 only from that term on.

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

Cf. A002997, A087788, A074379, A112428 - A112432, A006931.

ok[n_] :=  Divisible[n - 1, CarmichaelLambda[n]] && Length[FactorInteger[n]] > 3; Select[ Range[3*10^6], ok] (* Jean-François Alcover, Sep 23 2011 *)

A2997=readvec("b002997.gp"); A002997(n)=A2997[n]; for( n=1,100, omega( A002997(n) ) > 3 & print1( A002997(n)", "))

A141711 = A002997 \ A087788 = A074379 U A112428 U A112429 U A112430 U A112431 U A112432 U ...

A135721 a(n) is the smallest Carmichael number (A002997) divisible by the n-th prime, or 0 if no such number exists.

Original entry on oeis.org

561, 1105, 1729, 561, 1105, 561, 1729, 6601, 2465, 2821, 29341, 6601, 334153, 62745, 2433601, 74165065, 29341, 8911, 10024561, 10585, 2508013, 55462177, 62745, 46657, 101101, 52633, 84350561, 188461, 278545, 1152271, 18307381, 410041, 2628073, 12261061, 838201

Offset: 2

Artur Jasinski, Nov 25 2007

nonn

			561 is the first Carmichael number and its prime factors are 3, 11, 17 (2nd, 5th and 7th primes), so a(2), a(5) and a(7) are equal to 561. - _Michel Marcus_, Nov 07 2013

Amiram Eldar, Table of n, a(n) for n = 2..10001 (calculated using data from Claude Goutier; terms 2..1000 from Donovan Johnson)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Gérard P. Michon, Carmichael Multiples of Odd Primes.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A135717, A006931, A135719, A135720.

c = Cases[Range[1, 10000000, 2], n_ /; Mod[n, CarmichaelLambda@ n] == 1 && ! PrimeQ@ n]; Table[First@ Select[c, Mod[#, Prime@ n] == 0 &], {n, 2, 16}] (* Michael De Vlieger, Aug 28 2015, after Artur Jasinski at A002997 *)

Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
isA002997(n)=n%2 && !isprime(n) && Korselt(n) && n>1
a(n) = my(pn=prime(n),cn = 31*pn); until (isA002997(cn+=2*pn),); cn; \\ Michel Marcus, Nov 07 2013, improved by M. F. Hasler, Apr 14 2015

Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
a(n,p=prime(n))=my(m=lift(Mod(1/p,p-1)),c=max(m,33)*p,mp=m*p); while(!isprime(c) && !Korselt(c), c+=mp); c \\ Charles R Greathouse IV, Apr 15 2015

More terms from Michel Marcus, Nov 07 2013

Escape clause added by Jianing Song, Dec 12 2021

cp's OEIS Frontend

A135717 a(n) = number of prime divisors of Carmichael numbers A002997(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A174612 Number of four-prime Carmichael numbers less than 10^n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A174614 Number of six-prime Carmichael numbers less than 10^n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A174615 Number of seven-prime Carmichael numbers less than 10^n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A174616 Number of eight-prime Carmichael numbers less than 10^n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A174617 Number of nine-prime Carmichael numbers less than 10^n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A299711 Number of eleven-prime Carmichael numbers less than 10^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A112431 Carmichael numbers equal to the product of 8 primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A141711 Carmichael numbers with more than 3 prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs