A055553 -id:A055553 - cp's OEIS frontend

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

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

A258801 Carmichael numbers divisible by 3.

Original entry on oeis.org

561, 62745, 656601, 11921001, 26719701, 45318561, 174352641, 230996949, 662086041, 684106401, 689880801, 1534274841, 1848112761, 2176838049, 3022354401, 5860426881, 6025532241, 6097778961, 7281824001, 7397902401, 10031651841, 10054063041, 10585115841

Offset: 1

Fred Patrick Doty, Jun 10 2015

nonn

Most Carmichael numbers are congruent to 1 modulo 6. Those that are not are observed to include numbers that are 5 modulo 6 as well as multiples of 3.
Subsequence of A008585 and of A205947.
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.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..426, below 10^16, based on Richard Pinch data, from Giovanni Resta)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Richard G. E. Pinch, Tables relating to Carmichael numbers.
Index entries for sequences related to Carmichael numbers.

Cf. A002997 (Carmichael numbers), A205947 (Carmichael numbers not congruent to 1 modulo 6).
Cf. A008585 (3*n).
Cf. A045410 (primes not congruent to 1 modulo 6).

select(t -> t mod numtheory:-lambda(t) = 1, [seq(6*k+3,k=1..10^6)]); # Robert Israel, Jul 12 2015

Cases[Range[555,10^6,6],n_/;Mod[n,CarmichaelLambda[n]]==1]

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
is(n)=n%6==3 && Korselt(n) && n>9 \\ Charles R Greathouse IV, Jul 20 2015

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

A339909 Carmichael numbers k for which bigomega(phi(k)) < bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

Original entry on oeis.org

1729, 14676481, 84350561, 90698401, 279377281, 382536001, 413138881, 542497201, 702683101, 781347841, 851703301, 939947009, 955134181, 3480174001, 4765950001, 5255104513, 5781222721, 5985964801, 7558388641, 7816642561, 8714965001, 9237473281, 13630072501, 18189007201, 21669076801, 21863001601, 23915494401, 25477682491

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

Natural numbers n that satisfy equation k * phi(n) = n - 1, for some integer k > 1, should all occur in this sequence, if they exist at all. Lehmer conjectured that there are no such numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
Wikipedia, Lehmer's totient problem.
Index entries for sequences related to Carmichael numbers.

Intersection of A002997 and A339908.
Cf. A000010, A001222, A002322.
Cf. also A339818, A339869, A339878.

carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]]; Select[carmichaels, PrimeOmega[EulerPhi[#]] < PrimeOmega[# - 1] &] (* Amiram Eldar, Dec 26 2020 *)

A002322(n) = lcm(znstar(n)[2]); \\ From A002322
isA339909(n) = ((n>1)&&issquarefree(n)&&!isprime(n)&&(bigomega(eulerphi(n))A002322(n))));

A036060 Number of 3-component Carmichael numbers C = (6M + 1)(12M + 1)(18M + 1) < 10^n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 7, 10, 16, 25, 50, 86, 150, 256, 436, 783, 1435, 2631, 4765, 8766, 16320, 30601, 57719, 109504, 208822, 400643, 771735, 1494772, 2903761, 5658670, 11059937, 21696205, 42670184, 84144873, 66369603, 329733896, 655014986, 1303918824, 2601139051

Offset: 3

N. J. A. Sloane

nonn

Note that this is different from the count of 3-Carmichael numbers, A132195. The numbers counted here are neither those listed in A087788 (3 arbitrary prime factors) nor those listed in A033502 (where 6m + 1, 12m + 1 and 18m + 1 are all prime). - M. F. Hasler, Apr 14 2015

Posting by Harvey Dubner (harvey(AT)dubner.com) to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Nov 23 1998.

Amiram Eldar, Table of n, a(n) for n = 3..42
H. Dubner, 3-Component Carmichael Numbers-correction, Post to Number Theory List, Nov 23 1998.
Harvey Dubner, Carmichael Numbers of the form (6m+1)(12m+1)(18m+1), Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.1.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A033502, A055553, A087788, A132195.

Terms updated (from Dubner's paper) by Amiram Eldar, Aug 11 2017

A182490 Number of Carmichael numbers between 2^n and 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 1, 3, 1, 5, 4, 4, 10, 12, 10, 14, 26, 35, 32, 52, 76, 85, 108, 173, 208, 254, 328, 428, 563, 693, 928, 1130, 1454, 1879, 2481, 3234, 4164, 5231, 6890, 8855, 11309, 14905, 19227, 25040, 32035, 41615, 53710, 70061, 91228, 118940, 154659, 201004, 263363, 343053, 447613, 586096, 765319, 1000803, 1311065, 1716615, 2253877, 2956272, 3879379

Offset: 1

Brad Clardy, May 02 2012

nonn

While there may be an infinite number of Carmichael numbers, the ratio of Carmichael composites to odd composites (A094812), when looked at as a function of the power-of-two interval, apparently approaches 0 as the interval number n increases. It is 0.00533333 for n=10 but decreases to 0.00009035 by n=18 and is 0.00000254 at n=26, and looks like it could be reasonably modeled by 1/(A + B*log(n) + C*(log(n))^2 + D*(log(n)^3)).

Amiram Eldar, Table of n, a(n) for n = 1..72 (calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Richard Pinch, Carmichael numbers up to 10^21.

Cf. A002997.

for i:= 1 to 25 do
icount:=0;
for k := 2^i +1 to 2^(i+1)-1 by 2 do
  if (not IsPrime(k) and (k mod CarmichaelLambda(k) eq 1)) then icount +:=1;
  end if;
end for;
i, icount;
end for;

Extended to a(50) by T. D. Noe, May 02 2012

Extended to a(68) with data from R. Pinch by Brad Clardy, May 18 2014

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

Original entry on oeis.org

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.

A339878 Carmichael numbers k such that phi(k) divides p*(k - 1) for some prime factor p of k - 1.

Original entry on oeis.org

1729, 3069196417, 23915494401, 1334063001601, 6767608320001, 33812972024833, 1584348087168001, 1602991137369601, 6166793784729601, 1531757211193440001, 84388996672599528001

Offset: 1

Antti Karttunen (after Thomas Ordowski's and Amiram Eldar's SeqFan-posting), Dec 26 2020

The first ten terms are all in A339818, none is in A339869, and all except a(2) and a(6) are in A339909.

Also, for all ten, a(n) == 1 (mod 64). (Cf. a similar comment in A338998).

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Thomas Ordowski and Amiram Eldar, A new look at the Lehmer's totient problem, SeqFan, February 10 2019.

Intersection of A002997 and A338998.

Cf. also A339818, A339869, A339909.

carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]]; q[n_] := Module[{p = FactorInteger[n - 1][[;; , 1]], phi = EulerPhi[n]}, AnyTrue[(n - 1)*p, Divisible[#, phi] &]]; Select[carmichaels, q] (* Amiram Eldar, Dec 26 2020 *)

a(10) from Amiram Eldar, Dec 26 2020

a(11) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 21 2024

cp's OEIS Frontend

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

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

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

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A258801 Carmichael numbers divisible by 3.

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

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A339909 Carmichael numbers k for which bigomega(phi(k)) < bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

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A036060 Number of 3-component Carmichael numbers C = (6M + 1)(12M + 1)(18M + 1) < 10^n.

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A182490 Number of Carmichael numbers between 2^n and 2^(n+1).

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Magma

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