OFFSET
1,1
COMMENTS
Conjecture: k is a Carmichael number (A002997) if and only if k is a term of this sequence and all r-values of k are even.
From Ridouane Oudra, Apr 28 2019: (Start)
This sequence can also be defined as: Odd composite squarefree numbers k such that r' = 2*(k-1)/(p-1) is an integer for each prime divisor p of k. Proof:
2*(p - 2 + k/p)/(p-1) + 2*(k/p-1) = 2*(k-1)/(p-1),
so r is an integer if and only if r' is. (2*(k/p-1) is always an integer.)
With this new definition and Korselt's theorem it is easily shown that the proposed conjecture is true.
(End)
LINKS
K. Brockhaus, Table of n, a(n) for n = 1..653 (terms < 10^8)
PROG
(PARI) isok(n) = {if (((n % 2)==0) || isprime(n) || !issquarefree(n), return (0)); f = factor(n); for (i=1, #f~, d = f[i, 1]; if (type(2*(d-2+n/d)/(d-1)) != "t_INT", return(0)); ); return (1); } \\ Michel Marcus, Jul 12 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
William F. Sindelar (w_sindelar(AT)juno.com), Aug 19 2010
EXTENSIONS
Edited by the Associate Editors of the OEIS, Sep 04 2010
STATUS
approved