1,1

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

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

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).

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.

Counterexample: 46657 = (29341 + 63973)/2. - Charles R Greathouse IV, Dec 08 2014

Robert Israel, Table of n, a(n) for n = 1..10000

R. G. E. Pinch, Tables relating to Carmichael numbers

# Using data file from Richard Pinch

cfile:= "carmichael-16":

kor:= proc(t) local c;

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

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

end proc:

Res:= NULL: count:= 0:

while count < 100 do

S:= readline(cfile);

if S = 0 then break fi;

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

nL:= nops(L)-1;

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

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

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

fi;

od:

Res; # Robert Israel, Mar 20 2018

(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

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

Cf. A002997, A207041, A212920.

Sequence in context: A236873 A389427 A236608 * A265827 A212882 A290793

Adjacent sequences: A214755 A214756 A214757 * A214759 A214760 A214761

Marius Coman, Aug 03 2012

a(21) = 88689601 and a(26) = 133205761 inserted by Robert Israel, Mar 20 2018

approved