OFFSET
1,1
COMMENTS
The index of a Carmichael number k is i(k) = (k-1)/lambda(k).
Or, i(k) = (k-1)/lcm(p_1-1,p_2-1,...,p_j-1), where k = p_1*p_2*...*p_j. - Thomas Ordowski, Oct 15 2015
For composite k, lambda(k) divides k-1 iff k is a Carmichael number. - Thomas Ordowski, Oct 23 2015
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
J. M. Chick, Carmichael number variable relations: three-prime Carmichael numbers up to 10^24, arXiv:0711.2915 [math.NT] 2007-2008, Table 1, p. 34.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2002), no. 238, 883-908.
R. G. E. Pinch, Carmichael numbers up to 10^18, April 2006.
Richard Pinch, Carmichael Numbers up to 10^20, Abstract, ANTS 7.
Richard Pinch, Carmichael Numbers up to 10^20, Poster, ANTS 7.
FORMULA
EXAMPLE
a(1)= 7 because A002997(1) = 561, and (561 - 1)/lambda(561) = 560/80 = 7.
MAPLE
with(numtheory) : for n from 2 to 2000000 do: if type(n, prime)=false and issqrfree(n)=true then x:=factorset(n):n1:=nops(x):ii:=0:for j from 1 to n1 do:if irem(n-1, x[j]-1)=0 then ii:=ii+1:else fi:od:if ii=n1 then z:=(n-1)/lambda(n):printf(`%d, `, z):else fi:fi:od:
MATHEMATICA
carNums = Select[Range[561, 3 10^6, 2], CompositeQ[#] && Mod[#, CarmichaelLambda[#]] == 1&];
a[n_] := (carNums[[n]] - 1)/CarmichaelLambda[carNums[[n]]];
Array[a, 60] (* Jean-François Alcover, Sep 05 2018 *)
PROG
(PARI) t(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1;
for(n=1, 1e7, if(n%2 && !isprime(n) && t(n) && n>1, print1((n-1)/(lcm(znstar(n)[2])), ", "))) \\ Altug Alkan, Oct 15 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 23 2010, Mar 31 2010
EXTENSIONS
Edited by Michel Lagneau, Jul 31 2012
Further edits from N. J. A. Sloane, Oct 31 2015
STATUS
approved