A214758 - OEIS

%I #30 Apr 20 2023 10:48:24

%S 63973,126217,172081,188461,278545,748657,997633,1773289,5310721,

%T 8719921,8830801,9890881,15888313,18162001,26474581,26921089,31146661,

%U 36121345,37354465,41471521,88689601,93614521,93869665,101957401,120981601,133205761

%N Carmichael numbers divisible by a smaller Carmichael number.

%C Carmichael numbers by which the numbers from sequence are divisible: 1729, 1729, 2821, 1729, 2465, 1729, 1729, 8911, 29341, 6601, 8911, 41041, 8911, 75361, 8911, 46657, 2821 and 172081, 1105, 10585, 2821 and 172081, 41041, 41041, 2465 and 278545, 1729 and 188461, 46657, 552721.- Corrected by _Robert Israel_, Mar 20 2018

%C Note: A Carmichael number can be divisible by more than one Carmichael number: e.g. 31146661, 41471521, 101957401.

%C A subsequence of this sequence contains the numbers C1 (and another subsequence the numbers C3) that can be written as C1 = (C2 + C3)/2, where C1, C2 and C3 are Carmichael numbers and C1 and C3 are both divisible by C2 (e.g. 63973 = (1729 + 126217)/2; 93614521 = (41041 + 187188001)/2).

%C Conjecture: A Carmichael number C1 can be written as C1 = (C2 + C3)/2, where C2 and C3 are also Carmichael numbers, only if both C1 and C3 are divisible by C2.

%C Counterexample: 46657 = (29341 + 63973)/2. - _Charles R Greathouse IV_, Dec 08 2014

%H Robert Israel, <a href="/A214758/b214758.txt">Table of n, a(n) for n = 1..10000</a>

%H R. G. E. Pinch, <a href="http://www.s369624816.websitehome.co.uk/rgep/cartable.html">Tables relating to Carmichael numbers</a>

%p # Using data file from Richard Pinch

%p cfile:= "carmichael-16":

%p kor:= proc(t) local c;

%p    c:= convert(t,`*`);

%p    andmap(s -> c-1 mod (s-1) = 0, t)

%p end proc:

%p Res:= NULL: count:= 0:

%p while count < 100 do

%p     S:= readline(cfile);

%p     if S = 0 then break fi;

%p     L:= map(parse,StringTools:-Split(S));

%p     nL:= nops(L)-1;

%p     cands:= select(s -> nops(s) >= 3 and nops(s) < nL, combinat:-powerset(L[2..-1]));

%p     if select(kor, cands) <> [] then

%p         Res:= Res, L[1]; count:= count+1;

%p       fi;

%p od:

%p Res; # _Robert Israel_, Mar 20 2018

%o (PARI) Korselt(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,2]>1||(n-1)%(f[i,1]-1),return(0)));1

%o is(n)=if(n%2==0||isprime(n)||n<9||!Korselt(n), return(0)); fordiv(n,d,if(Korselt(d) && d>1 && !isprime(d), return(d<n))); 0 \\ _Charles R Greathouse IV_, Dec 08 2014

%Y Cf. A002997, A207041, A212920.

%K nonn

%O 1,1

%A _Marius Coman_, Aug 03 2012

%E a(21) = 88689601 and a(26) = 133205761 inserted by _Robert Israel_, Mar 20 2018