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Composite numbers m such that A309132(m) <= m.
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%I #27 Jul 19 2019 15:16:59

%S 561,1105,1729,2465,2821,5005,6601,8911,10585,15841,28405,29341,41041,

%T 46657,47125,52633,62745,63973,75361,98605,101101

%N Composite numbers m such that A309132(m) <= m.

%C It contains all the Carmichael numbers A002997 and the numbers 5005, 28405, 47125, 98605, ...

%C Carmichael numbers m for which A309132(m) < m are 561, 1105, 46657, 52633, ...

%C 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 Conjecture: Composite numbers m such that A309132(m) | m are only the Carmichael numbers. - _Amiram Eldar_ and _Thomas Ordowski_, Jul 18 2019

%C 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 f[n_] := Denominator[Numerator[BernoulliB[n - 1]] / n + Denominator[BernoulliB[n - 1]] / n^2]; Select[Range[10^4], CompositeQ[#] && f[#] <= # &]

%o (PARI) f(n) = my(b=bernfrac(n-1)); denominator(numerator(b)/n + denominator(b)/n^2); \\ A309132

%o isok(n) = (n>1) && !isprime(n) && (f(n) <= n); \\ _Michel Marcus_, Jul 17 2019

%Y Cf. A002997, A027641, A027642, A309132 (see the last conjecture).

%K nonn,hard,more

%O 1,1

%A _Amiram Eldar_ and _Thomas Ordowski_, Jul 17 2019