A002997 -id:A002997 - cp's OEIS frontend

A055553 Number of Carmichael numbers (A002997) less than 10^n.

0, 0, 1, 7, 16, 43, 105, 255, 646, 1547, 3605, 8241, 19279, 44706, 105212, 246683, 585355, 1401644, 3381806, 8220777, 20138200, 49679870

Offset: 1

Eric W. Weisstein

Richard Pinch, Carmichael Numbers up to 10^20, ANTS 7.
Richard G. E. Pinch, The Carmichael numbers up to 10^21, Proceedings of Conference on Algorithmic Number Theory 2007.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 219.

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]
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Claude Goutier, Compressed text file carm10e22.7z containing all the Carmichael numbers up to 10^22. [Local copy, with permission. This is a very large file.]
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 4, 2013.
Richard G. E. Pinch, The Carmichael numbers up to 10^17, arXiv:math/0504119 [math.NT], 2005.
Richard G. E. Pinch, The Carmichael numbers up to 10^18, arXiv:math/0604376 [math.NT], 2006.
Richard G. E. Pinch, Mathematics research page.
Andrew Shallue and Jonathan Webster, Advances in Tabulating Carmichael Numbers, Research in Number Theory, Vol. 11, No. 8 (2025), pp. 1-8; arXiv preprint, arXiv:2401.14495 [math.NT], 2024.
Eric Weisstein's World of Mathematics, Carmichael Number.
Eric Weisstein's World of Mathematics, Pseudoprime.

Cf. A002997.

Updates from Pinch's articles sent by Charles R Greathouse IV, Dec 04 2005, Jul 16 2006, May 29 2007
a(21) from Pinch's paper by Charles R Greathouse IV, Feb 01 2009
a(22) from Shallue and Webster (2024) added by Amiram Eldar, Feb 23 2024
a(22) = 49679870 reported by Claude Goutier on Dec 28 2022 (see links). - N. J. A. Sloane, Apr 18 2024

A135720 a(n) is the smallest Carmichael number (A002997) with the n-th prime as its smallest prime divisor, or 0 if no such number exists.

Original entry on oeis.org

561, 1105, 1729, 75361, 29341, 162401, 334153, 1615681, 3581761, 399001, 294409, 252601, 1152271, 104569501, 2508013, 178837201, 6189121, 10267951, 10024561, 14469841, 4461725581, 985052881, 19384289, 23382529, 3828001, 90698401, 84350561, 6733693, 17098369

Offset: 2

Artur Jasinski, Nov 25 2007

nonn

			a(2) = 561 because the smallest prime divisor of 561 is 3 which is the second prime.

Amiram Eldar, Table of n, a(n) for n = 2..1383 (calculated using data from Claude Goutier; terms 2..447 from Donovan Johnson, terms 448..615 from Max Alekseyev)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A135717, A006931, A135719, A135721, A141705.

Two missing terms and terms up to a(447) added by Donovan Johnson, Dec 25 2013
a(448)-a(615) in b-file from Max Alekseyev, Mar 11 2018
Escape clause added by Jianing Song, Dec 12 2021

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

Original entry on oeis.org

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

A153508 Sarrus numbers A001567 that are not Carmichael numbers A002997.

Original entry on oeis.org

341, 645, 1387, 1905, 2047, 2701, 3277, 4033, 4369, 4371, 4681, 5461, 7957, 8321, 8481, 10261, 11305, 12801, 13741, 13747, 13981, 14491, 15709, 16705, 18705, 18721, 19951, 23001, 23377, 25761, 30121, 30889, 31417, 31609, 31621, 33153, 34945

Offset: 1

Artur Jasinski, Dec 28 2008

nonn

A composite number n is a Fermat pseudoprime to base b if and only if b^(n-1) == 1 (mod n). Fermat pseudoprimes to base 2 are sometimes called Poulet numbers, Sarrus numbers, or frequently just pseudoprimes. For any given base pseudoprimes will contain Carmichael numbers as a subset. This sequence consists of base-2 Fermat pseudoprimes without the Carmichael numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..306 from Brad Clardy)

Cf. A001567, A002997.

for n:= 3 to 1052503 by 2 do
  if (IsOne(2^(n-1) mod n)
      and not IsPrime(n)
      and not n mod CarmichaelLambda(n) eq 1)
      then n;
      end if;
end for; // Brad Clardy, Dec 25 2014

filter:= proc(n)
local q;
   if isprime(n) then return false fi;
   if 2 &^(n-1) mod n <> 1 then return false fi;
   if not numtheory:-issqrfree(n) then return true fi;
   for q in numtheory:-factorset(n) do
     if (n-1) mod (q-1) <> 0 then return true fi;
   od:
   false
end proc:
select(filter, [$1..10^5]); # Robert Israel, Dec 29 2014

Select[Range[3, 35000, 2], !PrimeQ[#] && PowerMod[2, # - 1, # ] == 1 && !Divisible[# - 1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jun 25 2019 *)

A153513 Composite numbers k such that 2^k-2 and 3^k-3 are both divisible by k and k is not a Carmichael number (A002997).

Original entry on oeis.org

2701, 18721, 31621, 49141, 83333, 83665, 88561, 90751, 93961, 104653, 107185, 176149, 204001, 226801, 228241, 276013, 282133, 534061, 563473, 574561, 622909, 653333, 665281

Offset: 1

Artur Jasinski, Dec 28 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from Robert Israel)

Intersection of A153514 and A153508 (excluding the number 1).

Cf. A002997, A122780.

filter:= proc(n) local p;
  if isprime(n) or (2 &^n - 2 mod n <> 0) or (3 &^n - 3 mod n <> 0) then return false fi;
  if n::even then return true fi;
  if not numtheory:-issqrfree(n) then return true fi;
  for p in numtheory:-factorset(n) do
    if n-1 mod (p-1) <> 0 then return true fi
  od;
false
end proc:
select(filter, [$2..10^6]); # Robert Israel, Jan 29 2017

Reap[Do[If[CompositeQ[n] && Divisible[2^n-2, n] && Divisible[3^n-3, n] && Mod[n, CarmichaelLambda[n]] != 1, Print[n]; Sow[n]], {n, 2, 10^6}]][[2, 1]] (* Jean-François Alcover, Mar 25 2019 *)

A081702 Largest prime factor of the n-th Carmichael number (A002997).

Original entry on oeis.org

17, 17, 19, 29, 31, 41, 67, 73, 73, 61, 41, 97, 103, 89, 37, 31, 101, 241, 73, 233, 61, 109, 101, 113, 109, 397, 409, 67, 211, 137, 163, 181, 271, 421, 61, 197, 271, 199, 433, 73, 151, 61, 577, 271, 307, 37, 163, 211, 631, 541, 113, 353, 199, 331, 461, 101, 97

Offset: 1

Lekraj Beedassy, Apr 02 2003

nonn

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

Cf. A002997, A006530.

CarmichaelQ[n_] := Not[PrimeQ[n]] && Divisible[n - 1, CarmichaelLambda[n]]; FactorInteger[#][[-1, 1]]& /@ Select[Range[4, 10^7], CarmichaelQ] (* Amiram Eldar, Jun 24 2019 after Jean-François Alcover at A141710 *)

a(n) = A006530(A002997(n)). - Amiram Eldar, Jun 24 2019

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

A141710 Least prime factor of n-th Carmichael number A002997(n).

Original entry on oeis.org

3, 5, 7, 5, 7, 7, 7, 5, 7, 13, 7, 13, 7, 3, 7, 11, 7, 13, 7, 17, 7, 7, 41, 5, 37, 13, 19, 13, 31, 41, 5, 37, 31, 13, 13, 3, 11, 7, 7, 5, 7, 11, 7, 19, 7, 5, 7, 43, 31, 37, 17, 23, 7, 31, 41, 11, 19, 17, 13, 53, 5, 7, 11, 43, 13, 5, 29, 5, 101, 53, 7, 13, 11, 7, 19, 31, 41, 13, 29, 31, 5

Offset: 1

M. F. Hasler, Jul 01 2008

nonn

All terms of this sequence are odd primes. See A002997 for references.

			a(1)=3 since A002997(1)=3*11*17.

Amiram Eldar, Table of n, a(n) for n = 1..10000
OEIS index entries for Carmichael numbers

Cf. A002997, A020639, A081702, A135717.

CarmichaelQ[n_] := Not[PrimeQ[n]] && Divisible[n - 1, CarmichaelLambda[n]]; FactorInteger[#][[1, 1]]& /@ Select[Range[4, 10^7], CarmichaelQ] (* Jean-François Alcover, Sep 23 2011 *)

A141710(n)=vecmin(factor(A002997(n))[,1]) /* where A002997(n) can be defined as follows: */ system("wget b002997.txt; sed -e 's/^[0-9]*//' b002997.gp"); A2997=readvec("b002997.gp"); A002997(n)=A2997[n]; \

a(n) = A020639(A002997(n))

A153514 Terms of A122780 which are not Carmichael numbers A002997.

Original entry on oeis.org

1, 6, 66, 91, 121, 286, 671, 703, 726, 949, 1541, 1891, 2665, 2701, 3281, 3367, 3751, 4961, 5551, 7107, 7381, 8205, 8401, 8646, 11011, 12403, 14383, 15203, 15457, 16471, 16531, 18721, 19345, 23521, 24046, 24661, 24727, 28009, 29161, 30857, 31621

Offset: 1

Artur Jasinski, Dec 28 2008

nonn

For the intersection of this sequence and A153508, see A153513.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A002997, A122780, A153508, A153513.

filter:= proc(n) local p;
  if isprime(n) or (3 &^n - 3 mod n <> 0) then return false fi;
  if n::even then return true fi;
  if not numtheory:-issqrfree(n) then return true fi;
  for p in numtheory:-factorset(n) do
    if n-1 mod (p-1) <> 0 then return true fi
  od;
false
end proc:
filter(1):= true:
select(filter, [$1..10^5]); # Robert Israel, Jan 29 2017

okQ[n_] := !PrimeQ[n] && PowerMod[3, n, n] == Mod[3, n] && Mod[n, CarmichaelLambda[n]] != 1;
Select[Range[10^5], okQ] (* Jean-François Alcover, Mar 27 2019 *)

A213812 a(n) = smallest m for which the n-th Carmichael number A002997(n) can be written as p^2(m+1) - pm.

Original entry on oeis.org

1, 3, 4, 2, 2, 3, 1, 1, 2, 7, 24, 4, 4, 7, 47, 80, 9, 1, 23, 2, 46, 15, 24, 21, 24, 1, 1, 76, 8, 21, 16, 14, 6, 2, 150, 16, 8, 16, 3, 156, 36, 232, 2, 13, 10, 788, 40, 25, 2, 4, 123, 12, 44, 16, 8, 207, 226, 462, 92, 6

Offset: 1

Marius Coman, Jun 20 2012

nonn

The corresponding values of p are (we write the Carmichael number in brackets): 17(561), 17(1105), 19(1729), 29(2465), 31(2821), 41(6601), 67(8911), 73(10585), 73(15841), 61(29341), 41(41041), 97(46657), 103(52633), 89(62745), 37(63973), 31(75361), 101(101101), 241(115921), 73(126217), 233(162401), 61(172081), 109(188461), 101(252601), 113(278545), 109(294409), 397(314821), 409(334153), 67(340561), 211(399001), 137(410041), 163(449065), 181(488881), 271(512461), 421(530881), 61(552721), 197(656601), 271(658801), 199(670033), 433(748657), 73(825265), 151(838201), 61(852841), 577(997633), 271(1024651), 307(1033669), 37(1050985), 163(1082809), 211(1152271), 631(1193221), 541(1461241), 113(1569457), 353(1615681), 199(1773289), 331(1857241), 461(1909001), 101(2100901), 97(2113921), 73(2433601), 163(2455921), 599(2508013).
Any Carmichael number C can be written as C = p^2*(n+1) - p*n, where p is any prime divisor of C (it can be seen that the smallest n is obtained for the biggest prime divisor).
The formula C = p^2*(n+1) - p*n is equivalent to C = p^2*m - p*(m-1) = p^2*m - p*m + p, equivalent to p^2 - p divides C - p, which is a direct consequence of Korselt’s criterion.
It can be shown from p - 1 divides C - 1 not that just p^2 - p divides C - p but even that p^2 - p divides C - p^k (if C > p^k) or p^k - C (if p^k > C) which leads to the generic formula for a Carmichael number: C = p^k + n*p^2 - n*p (if C > p^k) or C = p^k - n*p^2 + n*p (if p^k > C) for any p prime divisor of C and any k natural number.
The formulas generated giving values of k seems to be very useful in the study of Fermat pseudoprimes; also, the composite numbers C for which the equation C = p^k - n*p^2 + n*p gives, over the integers, as solutions, all their prime divisors, for a certain k, deserve further study.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Carmichael Number

Cf. A002997.

Car=[561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217]; \\ use more terms of A002997 as desired
apply(C->my(f=factor(C)[,1],p=f[#f],p2=p^2); (C-p2)/(p2-p), Car) \\ Charles R Greathouse IV, Jul 05 2017

cp's OEIS Frontend

A055553 Number of Carmichael numbers (A002997) less than 10^n.

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A135720 a(n) is the smallest Carmichael number (A002997) with the n-th prime as its smallest prime divisor, or 0 if no such number exists.

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

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A153508 Sarrus numbers A001567 that are not Carmichael numbers A002997.

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A153513 Composite numbers k such that 2^k-2 and 3^k-3 are both divisible by k and k is not a Carmichael number (A002997).

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A081702 Largest prime factor of the n-th Carmichael number (A002997).

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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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A141710 Least prime factor of n-th Carmichael number A002997(n).

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A153514 Terms of A122780 which are not Carmichael numbers A002997.

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A213812 a(n) = smallest m for which the n-th Carmichael number A002997(n) can be written as p^2(m+1) - pm.