A180248 - OEIS

%I #15 May 03 2019 17:36:26

%S 15,91,435,561,703,1105,1729,1891,2465,2701,2821,3367,5551,6601,8695,

%T 8911,10585,11305,12403,13981,15051,15841,16471,18721,23001,26335,

%U 29341,30889,38503,39865,41041,46657,49141,52633,53131,62745

%N Odd composite squarefree numbers k such that r = 2*(p - 2 + k/p)/(p-1) is an integer for each prime divisor p of k.

%C 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.

%C From _Ridouane Oudra_, Apr 28 2019: (Start)

%C 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:

%C    2*(p - 2 + k/p)/(p-1) + 2*(k/p-1) = 2*(k-1)/(p-1),

%C so r is an integer if and only if r' is. (2*(k/p-1) is always an integer.)

%C With this new definition and Korselt's theorem it is easily shown that the proposed conjecture is true.

%C (End)

%H K. Brockhaus, <a href="/A180248/b180248.txt">Table of n, a(n) for n = 1..653</a> (terms < 10^8)

%o (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

%K nonn

%O 1,1

%A William F. Sindelar (w_sindelar(AT)juno.com), Aug 19 2010

%E Edited by the Associate Editors of the OEIS, Sep 04 2010