The OEIS is supported by the many generous donors to the OEIS Foundation.
Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #45 Sep 30 2023 02:05:17
%S 1729,12801,5079361,34479361,3069196417,23915494401
%N Composite numbers k such that phi(k) divides p*(k - 1) for some prime factor p of k - 1.
%C All terms of this sequence are terms of A337316 and all Lehmer numbers (if there are any) are contained in this sequence.
%C Terms 1729 and 3069196417 and several others are also Carmichael numbers (A002997), they are given in A339878.
%C The sequence also includes: 1334063001601, 6767608320001, 33812972024833, 380655711289345, 1584348087168001, 1602991137369601, 6166793784729601, 1531757211193440001. - _Daniel Suteu_, Nov 24 2020
%C Apparently, a(n) == 1 (mod 64). - _Hugo Pfoertner_, Dec 08 2020
%C The "Lehmer numbers" above refers to composite 1-Lehmer numbers, that is, numbers n that would satisfy the equation y * phi(n) = n-1, for some integer y > 1. Lehmer conjectured that no such numbers exist. See the assorted Web-links. - _Antti Karttunen_, Dec 26 2020
%H D. H. Lehmer, <a href="https://doi.org/10.1090/S0002-9904-1932-05521-5">On Euler's totient function</a>, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
%H Giovanni Resta, <a href="https://www.numbersaplenty.com/set/Lehmer_number/">k-Lehmer numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LehmersTotientProblem.html">Lehmer's Totient Problem</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer's_totient_problem">Lehmer's totient problem</a>
%e phi(1729) = 1296 divides 3 * 1728.
%o (PARI) is(n)={my(s=denominator((n-1)/eulerphi(n))); !isprime(n) && isprime(s) && ((n-1)%s==0) && n>1}
%o { forcomposite(n=1, 2^32, if(is(n), print1(n, ", "))) }
%Y Subsequence of A173703 (2-Lehmer numbers).
%Y Cf. A337316 (with "squarefree divisor" instead of "prime factor").
%Y Cf. A000010 (phi), A238574 (k-Lehmer numbers for some k), A339878 (Carmichael numbers in this sequence).
%K nonn,more
%O 1,1
%A _Tomohiro Yamada_, Nov 18 2020
%E a(5) from _Amiram Eldar_, Nov 18 2020
%E a(6) from _Daniel Suteu_, confirmed by _Max Alekseyev_, Sep 29 2023