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A124240 Numbers n such that lambda(n) divides n, where lambda is Carmichael's function (A002322). 26

%I #75 Dec 27 2023 11:55:37

%S 1,2,4,6,8,12,16,18,20,24,32,36,40,42,48,54,60,64,72,80,84,96,100,108,

%T 120,126,128,144,156,160,162,168,180,192,200,216,220,240,252,256,272,

%U 288,294,300,312,320,324,336,342,360,378,384,400,420,432,440,468,480

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

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

%C 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.

%C 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.

%C Also, the sequence of numbers n such that p-1 divides n for all primes p that divide n. - _Leroy Quet_, Jun 27 2008

%C Numbers n such that b^n == 1 (mod n) for every b coprime to n. - _Thomas Ordowski_, Jun 23 2017

%C Numbers m such that every divisor < m is the difference between two divisors of m. - _Michel Lagneau_, Aug 11 2017

%C 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

%H Alois P. Heinz, <a href="/A124240/b124240.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%H Alexander Kalmynin, <a href="https://arxiv.org/abs/1611.00417">On Novák numbers</a>, arXiv:1611.00417 [math.NT], 2016. See Theorem 6 p. 11 where these numbers are called Novák-Carmichael numbers.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarmichaelFunction.html">Carmichael Function</a>

%F k is in a <=> Clausen(k, 0) divides Clausen(k, 1), (Clausen = A160014). - _Peter Luschny_, Jun 08 2023

%e a(1) = 1 because 1 divides A124239(1) = 1.

%e a(2) = 2 because 2 divides A124239(2) = 14.

%e a(3) = 4 because 4 divides A124239(4) = 3704, but 3 does not divide A124239(3) = 197.

%p a:= proc(n) option remember; local k;

%p for k from `if`(n=1, 0, a(n-1))+1 while

%p irem(k, numtheory[lambda](k))>0 do od: k

%p end:

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jul 04 2021

%p # Using function 'Clausen' from A160014:

%p aList := m -> select(k -> irem(Clausen(k, 1), Clausen(k, 0)) = 0, [seq(1..m)]):

%p aList(480); # _Peter Luschny_, Jun 08 2023

%t 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}]

%t Flatten[Position[Table[n/CarmichaelLambda[n], {n, 440}], _Integer]] (* _T. D. Noe_, Sep 11 2008 *)

%o (Haskell)

%o a124240 n = a124240_list !! (n-1)

%o a124240_list = filter

%o (\x -> all (== 0) $ map ((mod x) . pred) $ a027748_row x) [1..]

%o -- _Reinhard Zumkeller_, Aug 27 2013

%o (PARI) is(n)=n%lcm(znstar(n)[2])==0 \\ _Charles R Greathouse IV_, Feb 11 2015

%o (Python)

%o from itertools import islice, count

%o from sympy.ntheory.factor_ import reduced_totient

%o def A124240gen(): return filter(lambda n:n % reduced_totient(n) == 0,count(1))

%o A124240_list = list(islice(A124240gen(),20)) # _Chai Wah Wu_, Dec 11 2021

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

%K nonn

%O 1,2

%A _Alexander Adamchuk_, Oct 22 2006

%E New definition from _T. D. Noe_, Aug 31 2008

%E Edited by _Max Alekseyev_, Aug 25 2013

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)