login
Composite numbers k such that (Bell(k+1) - Bell(k)) mod k = 1.
0

%I #27 Jul 12 2021 03:32:24

%S 4,28,40,343,10744,18506,18658,22360,34486,289912,293710,565213,

%T 722765,2469287,13231942,86523219

%N Composite numbers k such that (Bell(k+1) - Bell(k)) mod k = 1.

%C The congruence is true for all primes k. Bell(k) is the sequence A000110. Tested up to k=5000.

%C a(10) > 73000. - _Giovanni Resta_, Aug 26 2018

%C a(17) > 10^8. - _Hiroaki Yamanouchi_, Sep 01 2018

%C One could compute the Bell numbers mod lcm(1, 2, ..., k) (see A003418) (or even the lcm of the composite numbers up to k) to reduce the number of digits and still find the same terms. - _David A. Corneth_, Aug 26 2018

%e For k=4, (Bell(5) - Bell(4)) mod 4 = (52 - 15) mod 4 = 37 mod 4 = 1, but 4 is not prime, so 4 is a term.

%t fQ[n_] := ! PrimeQ@n && Mod[BellB[n + 1] - BellB[n], n] == 1; k = 1; lst = {}; While[k < 9201, If[fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* _Robert G. Wilson v_, Jul 28 2010 *)

%Y Cf. A000110, A003418.

%K nonn,more

%O 1,1

%A _Jean-Claude Arbaut_, Jul 08 2010

%E a(5)-a(9) from _Giovanni Resta_, Aug 26 2018

%E a(10)-a(16) from _Hiroaki Yamanouchi_, Sep 01 2018