login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A006931 Least Carmichael number with n prime factors, or 0 if no such number exists.
(Formerly M5463)
29
561, 41041, 825265, 321197185, 5394826801, 232250619601, 9746347772161, 1436697831295441, 60977817398996785, 7156857700403137441, 1791562810662585767521, 87674969936234821377601, 6553130926752006031481761, 1590231231043178376951698401 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
COMMENTS
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
REFERENCES
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).
LINKS
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
MATHEMATICA
(* 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 *)
PROG
(PARI) 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
(PARI)
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
CROSSREFS
Sequence in context: A264214 A083736 A182090 * A258801 A329460 A097061
KEYWORD
nonn
AUTHOR
EXTENSIONS
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
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)