A214758 Carmichael numbers divisible by a smaller Carmichael number.

63973, 126217, 172081, 188461, 278545, 748657, 997633, 1773289, 5310721, 8719921, 8830801, 9890881, 15888313, 18162001, 26474581, 26921089, 31146661, 36121345, 37354465, 41471521, 88689601, 93614521, 93869665, 101957401, 120981601, 133205761

Offset

1,1

Comments

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

Links

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

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

Maple

# 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

Prog

(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

Crossrefs

Cf. A002997, A207041, A212920.

Sequence in context: A145437 A236873 A236608 * A265827 A212882 A290793

Adjacent sequences: A214755 A214756 A214757 * A214759 A214760 A214761

Keyword

nonn

Author

Marius Coman, Aug 03 2012

Extensions

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

Status

approved