A174590 a(n) = (k-1)/lambda(k), the index of the n-th Carmichael number k.

7, 23, 48, 22, 47, 5, 45, 21, 44, 163, 342, 162, 43, 31, 1777, 314, 337, 161, 1753, 70, 2868, 1745, 421, 2487, 1363, 159, 39, 645, 950, 67, 198, 1358, 949, 158, 2303, 134, 305, 1692, 1733, 5731, 2794, 7107, 1732, 345, 1689, 2654, 1671, 1829, 947, 1353, 1557

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

a(n) = (A002997(n) - 1) / lambda(A002997(n)).

a(n) = (A002997(n) - 1) / A306414(n). - Jianing Song, Dec 12 2021

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

Cf. A002322 (the Carmichael lambda function), A002997, A011773, A306414.

Sequence in context: A185955 A158035 A101789 * A162290 A180044 A062725

Adjacent sequences: A174587 A174588 A174589 * A174591 A174592 A174593

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