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A338998 Composite numbers k such that phi(k) divides p*(k - 1) for some prime factor p of k - 1. 1
1729, 12801, 5079361, 34479361, 3069196417, 23915494401 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All terms of this sequence are terms of A337316 and all Lehmer numbers (if there are any) are contained in this sequence.
Terms 1729 and 3069196417 and several others are also Carmichael numbers (A002997), they are given in A339878.
The sequence also includes: 1334063001601, 6767608320001, 33812972024833, 380655711289345, 1584348087168001, 1602991137369601, 6166793784729601, 1531757211193440001. - Daniel Suteu, Nov 24 2020
Apparently, a(n) == 1 (mod 64). - Hugo Pfoertner, Dec 08 2020
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
LINKS
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
Giovanni Resta, k-Lehmer numbers
Eric Weisstein's World of Mathematics, Lehmer's Totient Problem
EXAMPLE
phi(1729) = 1296 divides 3 * 1728.
PROG
(PARI) is(n)={my(s=denominator((n-1)/eulerphi(n))); !isprime(n) && isprime(s) && ((n-1)%s==0) && n>1}
{ forcomposite(n=1, 2^32, if(is(n), print1(n, ", "))) }
CROSSREFS
Subsequence of A173703 (2-Lehmer numbers).
Cf. A337316 (with "squarefree divisor" instead of "prime factor").
Cf. A000010 (phi), A238574 (k-Lehmer numbers for some k), A339878 (Carmichael numbers in this sequence).
Sequence in context: A154728 A286217 A337316 * A280428 A194263 A339818
KEYWORD
nonn,more
AUTHOR
Tomohiro Yamada, Nov 18 2020
EXTENSIONS
a(5) from Amiram Eldar, Nov 18 2020
a(6) from Daniel Suteu, confirmed by Max Alekseyev, Sep 29 2023
STATUS
approved

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)