A339869 Carmichael numbers k for which A053575(k) [the odd part of phi] divides k-1.

561, 1105, 2465, 6601, 8911, 10585, 46657, 62745, 162401, 410041, 449065, 5148001, 5632705, 6313681, 6840001, 7207201, 11119105, 11921001, 19683001, 21584305, 26719701, 41298985, 55462177, 64774081, 67371265, 79411201, 83966401, 87318001, 99861985, 100427041, 172290241, 189941761, 484662529, 790623289, 809883361

Offset

1,1

Comments

Lehmer conjectured that the equation k * phi(n) = n - 1 (with k integer) has no solutions for any composite n (i.e., when k > 1). If this sequence has no common terms with A339818, then the conjecture certainly holds.

Links

Amiram Eldar, Table of n, a(n) for n = 1..1150 (terms below 10^22 calculated using data from Claude Goutier)

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.

Wikipedia, Lehmer's totient problem.

Index entries for sequences related to Carmichael numbers.

Mathematica

carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {_, _}][[;; , 2]]; oddPart[n_] := n/2^IntegerExponent[n, 2]; q[n_] := Divisible[n - 1, oddPart[EulerPhi[n]]]; Select[carmichaels, q] (* Amiram Eldar, Dec 26 2020 *)

Prog

(PARI)
A000265(n) = (n>>valuation(n, 2));
A002322(n) = lcm(znstar(n)[2]);
isA339869(n) = ((n>1)&&!isprime(n)&&(!((n-1)%A002322(n)))&&!((n-1)%A000265(eulerphi(n))));

Crossrefs

Intersection of A002997 and A339880.

Complement of A340092 in A002997.

Cf. A000010, A000265, A002322, A053575.

Cf. also A339818, A339878, A339909.

Sequence in context: A135720 A263403 A083733 * A375875 A214428 A262043

Adjacent sequences: A339866 A339867 A339868 * A339870 A339871 A339872

Keyword

nonn

Author

Antti Karttunen, Dec 22 2020

Status

approved