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

%I A174590 #44 Dec 12 2021 20:25:29
%S A174590 7,23,48,22,47,5,45,21,44,163,342,162,43,31,1777,314,337,161,1753,70,
%T A174590 2868,1745,421,2487,1363,159,39,645,950,67,198,1358,949,158,2303,134,
%U A174590 305,1692,1733,5731,2794,7107,1732,345,1689,2654,1671,1829,947,1353,1557
%N A174590 a(n) = (k-1)/lambda(k), the index of the n-th Carmichael number k.
%C A174590 The index of a Carmichael number k is i(k) = (k-1)/lambda(k).
%C A174590 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
%C A174590 For composite k, lambda(k) divides k-1 iff k is a Carmichael number. - _Thomas Ordowski_, Oct 23 2015
%H A174590 Robert Israel, <a href="/A174590/b174590.txt">Table of n, a(n) for n = 1..10000</a>
%H A174590 J. M. Chick, <a href="http://arxiv.org/abs/0711.2915">Carmichael number variable relations: three-prime Carmichael numbers up to 10^24</a>, arXiv:0711.2915 [math.NT] 2007-2008, Table 1, p. 34.
%H A174590 Andrew Granville and Carl Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/PDF/paper125.pdf">Two contradictory conjectures concerning Carmichael numbers</a>, Math. Comp. 71 (2002), no. 238, 883-908.
%H A174590 R. G. E. Pinch, <a href="http://www.chalcedon.demon.co.uk/rgep/publish.html#76">Carmichael numbers up to 10^18</a>, April 2006.
%H A174590 Richard Pinch, <a href="http://page.math.tu-berlin.de/~kant/ants/Poster/Pinch_1_Abstract.pdf">Carmichael Numbers up to 10^20</a>, Abstract, ANTS 7.
%H A174590 Richard Pinch, <a href="http://page.math.tu-berlin.de/~kant/ants/Poster/Pinch_Poster4.pdf">Carmichael Numbers up to 10^20</a>, Poster, ANTS 7.
%F A174590 a(n) = (A002997(n) - 1) / lambda(A002997(n)).
%F A174590 a(n) = (A002997(n) - 1) / A306414(n). - _Jianing Song_, Dec 12 2021
%e A174590 a(1)= 7 because A002997(1) = 561, and (561 - 1)/lambda(561) = 560/80 = 7.
%p A174590 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:
%t A174590 carNums = Select[Range[561, 3 10^6, 2], CompositeQ[#] && Mod[#, CarmichaelLambda[#]] == 1&];
%t A174590 a[n_] := (carNums[[n]] - 1)/CarmichaelLambda[carNums[[n]]];
%t A174590 Array[a, 60] (* _Jean-François Alcover_, Sep 05 2018 *)
%o A174590 (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;
%o A174590 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
%Y A174590 Cf. A002322 (the Carmichael lambda function), A002997, A011773, A306414.
%K A174590 nonn
%O A174590 1,1
%A A174590 _Michel Lagneau_, Mar 23 2010, Mar 31 2010
%E A174590 Edited by _Michel Lagneau_, Jul 31 2012
%E A174590 Further edits from _N. J. A. Sloane_, Oct 31 2015