cp's OEIS Frontend

A309235 Composite numbers m such that A309132(m) <= m.

%I A309235 #27 Jul 19 2019 15:16:59
%S A309235 561,1105,1729,2465,2821,5005,6601,8911,10585,15841,28405,29341,41041,
%T A309235 46657,47125,52633,62745,63973,75361,98605,101101
%N A309235 Composite numbers m such that A309132(m) <= m.
%C A309235 It contains all the Carmichael numbers A002997 and the numbers 5005, 28405, 47125, 98605, ...
%C A309235 Carmichael numbers m for which A309132(m) < m are 561, 1105, 46657, 52633, ...
%C A309235 If m is a Carmichael number, then not only is A309132(m) <= m, but in fact A309132(m) | m. (Proof. As m is a Carmichael number, m | D(m-1) where B(k) = N(k)/D(k) is the k-th Bernoulli number. So I := N(m-1) + D(m-1)/m is an integer. Hence A309132(m) = Denominator(I/m) is a divisor of m.) - _Jonathan Sondow_, Jul 17 2019
%C A309235 Conjecture: Composite numbers m such that A309132(m) | m are only the Carmichael numbers. - _Amiram Eldar_ and _Thomas Ordowski_, Jul 18 2019
%C A309235 If A309132(m) | m and m | A027642(m-1), then A309132(m) | A027642(m-1). It seems that, according to the data, a composite m is a Carmichael number if and only if A309132(m) | A027642(m-1). - _Thomas Ordowski_, Jul 19 2019
%t A309235 f[n_] := Denominator[Numerator[BernoulliB[n - 1]] / n + Denominator[BernoulliB[n - 1]] / n^2]; Select[Range[10^4], CompositeQ[#] && f[#] <= # &]
%o A309235 (PARI) f(n) = my(b=bernfrac(n-1)); denominator(numerator(b)/n + denominator(b)/n^2); \\ A309132
%o A309235 isok(n) = (n>1) && !isprime(n) && (f(n) <= n); \\ _Michel Marcus_, Jul 17 2019
%Y A309235 Cf. A002997, A027641, A027642, A309132 (see the last conjecture).
%K A309235 nonn,hard,more
%O A309235 1,1
%A A309235 _Amiram Eldar_ and _Thomas Ordowski_, Jul 17 2019