A124240 Numbers n such that lambda(n) divides n, where lambda is Carmichael's function (A002322).

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 42, 48, 54, 60, 64, 72, 80, 84, 96, 100, 108, 120, 126, 128, 144, 156, 160, 162, 168, 180, 192, 200, 216, 220, 240, 252, 256, 272, 288, 294, 300, 312, 320, 324, 336, 342, 360, 378, 384, 400, 420, 432, 440, 468, 480

Offset

1,2

Comments

Numbers n such that A124239(n) is divisible by n.

If k is in the sequence then 2k is also in the sequence, but if 2m is in the sequence m is not necessarily a term of the sequence.

This sequence is a subsequence of A068563. The first term that is different is A068563(27) = 136. The terms of A068563 that are not the terms of a(n) are listed in A124241.

Also, the sequence of numbers n such that p-1 divides n for all primes p that divide n. - Leroy Quet, Jun 27 2008

Numbers n such that b^n == 1 (mod n) for every b coprime to n. - Thomas Ordowski, Jun 23 2017

Numbers m such that every divisor < m is the difference between two divisors of m. - Michel Lagneau, Aug 11 2017

All terms > 1 in this sequence are even. Furthermore, either 4 or 6 divides a(n) for n > 3. 1806 is the largest squarefree term. - Paul Vanderveen, Apr 24 2022

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Alexander Kalmynin, On Novák numbers, arXiv:1611.00417 [math.NT], 2016. See Theorem 6 p. 11 where these numbers are called Novák-Carmichael numbers.

Eric Weisstein's World of Mathematics, Carmichael Function

Formula

k is in a <=> Clausen(k, 0) divides Clausen(k, 1), (Clausen = A160014). - Peter Luschny, Jun 08 2023

Example

a(1) = 1 because 1 divides A124239(1) = 1.
a(2) = 2 because 2 divides A124239(2) = 14.
a(3) = 4 because 4 divides A124239(4) = 3704, but 3 does not divide A124239(3) = 197.

Maple

a:= proc(n) option remember; local k;
       for k from `if`(n=1, 0, a(n-1))+1 while
       irem(k, numtheory[lambda](k))>0 do od: k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 04 2021
# Using function 'Clausen' from A160014:
aList := m -> select(k -> irem(Clausen(k, 1), Clausen(k, 0)) = 0, [seq(1..m)]):
aList(480); # Peter Luschny, Jun 08 2023

Mathematica

Do[f=n + Sum[ (2k-1)((2k-1)^n-1) / (2(k-1)), {k, 2, n} ]; If[IntegerQ[f/n], Print[n]], {n, 1, 900}]
Flatten[Position[Table[n/CarmichaelLambda[n], {n, 440}], _Integer]] (* T. D. Noe, Sep 11 2008 *)

Prog

(Haskell)
a124240 n = a124240_list !! (n-1)
a124240_list = filter
   (\x -> all (== 0) $ map ((mod x) . pred) $ a027748_row x) [1..]
-- Reinhard Zumkeller, Aug 27 2013
(PARI) is(n)=n%lcm(znstar(n)[2])==0 \\ Charles R Greathouse IV, Feb 11 2015
(Python)
from itertools import islice, count
from sympy.ntheory.factor_ import reduced_totient
def A124240gen(): return filter(lambda n:n % reduced_totient(n) == 0, count(1))
A124240_list = list(islice(A124240gen(), 20)) # Chai Wah Wu, Dec 11 2021

Crossrefs

Cf. A002322, A124239, A124241, A068563, A027748, A140470, A141766, A160014, A363523.

Sequence in context: A177807 A305726 A068563 * A320580 A325763 A363949

Adjacent sequences: A124237 A124238 A124239 * A124241 A124242 A124243

Keyword

nonn

Author

Alexander Adamchuk, Oct 22 2006

Extensions

New definition from T. D. Noe, Aug 31 2008

Edited by Max Alekseyev, Aug 25 2013

Status

approved