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A290281 Numbers k such that (k-1) mod phi(k) = lambda(k), where phi = A000010 and lambda = A002322.

%I A290281 #53 Apr 22 2024 14:28:39
%S A290281 6601,11972017,34657141,67902031,139952671,258634741,2000436751,
%T A290281 8801128801,9116583841,9462932431,38069223721,326170416001,
%U A290281 359316634951,1860929324101,2022188518351,2283475947391,2648686458601,2697891108151,4513362899761,5020030521001,5472940991761,6163867710001,7507903975951,19288340548471
%N A290281 Numbers k such that (k-1) mod phi(k) = lambda(k), where phi = A000010 and lambda = A002322.
%C A290281 Numbers k such that A215486(k) = A002322(k).
%C A290281 Subsequence of the Carmichael numbers (A002997).
%C A290281 Composite numbers k such that (k-1) == lambda(k) (mod phi(k)).
%C A290281 Composite numbers k such that A277127(k) == 1 (mod A000010(k)).
%C A290281 Problem: are there infinitely many such numbers?
%C A290281 Conjecture: these are numbers k such that phi(k) + lambda(k) = k - 1. Checked up to 2^64. - _Amiram Eldar_ and _Thomas Ordowski_, Dec 06 2019
%H A290281 Amiram Eldar, <a href="/A290281/b290281.txt">Table of n, a(n) for n = 1..239</a> (terms below 10^22 calculated using data from Claude Goutier; terms 1..79 from Robert Israel)
%H A290281 Claude Goutier, <a href="http://www-labs.iro.umontreal.ca/~goutier/OEIS/A055553/">Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22</a>.
%H A290281 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%p A290281 # Using data files for A002997
%p A290281 count:= 0:
%p A290281 for cfile in ["carmichael-16","carmichael17","carmichael18"] do
%p A290281 do
%p A290281     S:= readline(cfile);
%p A290281     if S = 0 then break fi;
%p A290281     L:= map(parse, StringTools:-Split(S));
%p A290281     n:= L[1]; pm:= map(`-`,L[2..-1],1);
%p A290281     phin:= convert(pm,`*`);
%p A290281     lambdan:= ilcm(op(pm));
%p A290281     if n-1 - lambdan mod phin = 0 then
%p A290281       count:= count+1; A[count]:= n;
%p A290281     fi
%p A290281 od:
%p A290281    fclose(cfile);
%p A290281 od:
%p A290281 seq(A[i],i=1..count); # _Robert Israel_, Jul 26 2017
%t A290281 Select[Range[10^8], Divisible[# - 1, (lam = CarmichaelLambda[#])] && Mod[# - 1, EulerPhi[#]] == lam &] (* _Amiram Eldar_, Dec 06 2019 *)
%Y A290281 Cf. A000010, A002322, A002997, A215486.
%Y A290281 Subsequence of A264012.
%K A290281 nonn
%O A290281 1,1
%A A290281 _Robert Israel_ and _Thomas Ordowski_, Jul 25 2017