A135719 -id:A135719 - cp's OEIS frontend

A006931 Least Carmichael number with n prime factors, or 0 if no such number exists.

561, 41041, 825265, 321197185, 5394826801, 232250619601, 9746347772161, 1436697831295441, 60977817398996785, 7156857700403137441, 1791562810662585767521, 87674969936234821377601, 6553130926752006031481761, 1590231231043178376951698401

Offset: 3

N. J. A. Sloane and Richard Pinch

nonn

Alford, Grantham, Hayman, & Shallue construct large Carmichael numbers, finding upper bounds for a(3)-a(19565220) and a(10333229505). - Charles R Greathouse IV, May 30 2013

J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 269, Pour la Science, Paris 2000.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 3..35
W. R. Alford, Jon Grantham, Steven Hayman, Andrew Shallue, Constructing Carmichael numbers through improved subset-product algorithms, arXiv:1203.6664 [math.NT], 2012-2013.
R. G. E. Pinch, The Carmichael numbers up to 10^15, Math. Comp. 61 (1993), no. 203, 381-391.
R. G. E. Pinch, The Carmichael numbers up to 10^17, arXiv:math/0504119 [math.NT], 2005.
R. G. E. Pinch, The Carmichael numbers up to 10^18, arXiv:math/0604376 [math.NT], 2006.
Eric Weisstein's World of Mathematics, Carmichael Number
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A135717, A135719, A135720, A135721.

Cf. A087788, A141711, A074379, A112428, A112429, A112430, A112431, A112432.

(* Program not suitable to compute more than a few terms *)
A2997 = Select[Range[1, 10^6, 2], CompositeQ[#] && Mod[#, CarmichaelLambda[#] ] == 1&];
(First /@ Split[Sort[{PrimeOmega[#], #}& /@ A2997], #1[[1]] == #2[[1]]&])[[All, 2]] (* Jean-François Alcover, Sep 11 2018 *)

Korselt(n,f=factor(n))=for(i=1,#f[,1],if(f[i,2]>1||(n-1)%(f[i,1]-1),return(0)));1
a(n)=my(p=2,f);forprime(q=3,default(primelimit),forstep(k=p+2,q-2,2,f=factor(k);if(vecmax(f[,2])==1 && #f[,2]==n && Korselt(k,f), return(k)));p=q)
\\ Charles R Greathouse IV, Apr 25 2012

carmichael(A, B, k) = A=max(A, vecprod(primes(k+1))\2); (f(m, l, lo, k) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, lo=max(lo, ceil(A/m)); my(t=lift(1/Mod(m,l))); while(t < lo, t += l); forstep(p=t, hi, l, if(isprime(p), my(n=m*p); if((n-1)%(p-1) == 0, listput(list, n)))), forprime(p=lo, hi, if(gcd(m, p-1) == 1, list=concat(list, f(m*p, lcm(l, p-1), p+1, k-1))))); list); vecsort(Vec(f(1, 1, 3, k)));
a(n) = if(n < 3, return()); my(x=vecprod(primes(n+1))\2,y=2*x); while(1, my(v=carmichael(x,y,n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 24 2023

Corrected by Lekraj Beedassy, Dec 31 2002
More terms from Ralf Stephan, from the Pinch paper, Apr 16 2005
Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar.
Escape clause added by Jianing Song, Dec 12 2021

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

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

A006931 Least Carmichael number with n prime factors, or 0 if no such number exists.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions