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

6601, 11972017, 34657141, 67902031, 139952671, 258634741, 2000436751, 8801128801, 9116583841, 9462932431, 38069223721, 326170416001, 359316634951, 1860929324101, 2022188518351, 2283475947391, 2648686458601, 2697891108151, 4513362899761, 5020030521001, 5472940991761, 6163867710001, 7507903975951, 19288340548471

Offset: 1

Author

Robert Israel and Thomas Ordowski, Jul 25 2017

Keywords

• nonn

Comments

Numbers k such that A215486(k) = A002322(k).

Subsequence of the Carmichael numbers (A002997).

Composite numbers k such that (k-1) == lambda(k) (mod phi(k)).

Composite numbers k such that A277127(k) == 1 (mod A000010(k)).

Problem: are there infinitely many such numbers?

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..239 (terms below 10^22 calculated using data from Claude Goutier; terms 1..79 from Robert Israel)
• Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Cf. A000010, A002322, A002997, A215486.

Subsequence of A264012.

Programs

Maple

# Using data files for A002997
count:= 0:
for cfile in ["carmichael-16","carmichael17","carmichael18"] do
do
    S:= readline(cfile);
    if S = 0 then break fi;
    L:= map(parse, StringTools:-Split(S));
    n:= L[1]; pm:= map(`-`,L[2..-1],1);
    phin:= convert(pm,`*`);
    lambdan:= ilcm(op(pm));
    if n-1 - lambdan mod phin = 0 then
      count:= count+1; A[count]:= n;
    fi
od:
   fclose(cfile);
od:
seq(A[i],i=1..count); # Robert Israel, Jul 26 2017

Mathematica

Select[Range[10^8], Divisible[# - 1, (lam = CarmichaelLambda[#])] && Mod[# - 1, EulerPhi[#]] == lam &] (* Amiram Eldar, Dec 06 2019 *)