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

561, 1105, 1729, 2465, 2821, 5005, 6601, 8911, 10585, 15841, 28405, 29341, 41041, 46657, 47125, 52633, 62745, 63973, 75361, 98605, 101101

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, Jul 17 2019

Keywords

• nonn
• hard
• more

Comments

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

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

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

Conjecture: Composite numbers m such that A309132(m) | m are only the Carmichael numbers. - Amiram Eldar and Thomas Ordowski, Jul 18 2019

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

Crossrefs

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

Programs

Mathematica

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

PARI

f(n) = my(b=bernfrac(n-1)); denominator(numerator(b)/n + denominator(b)/n^2); \\ A309132
isok(n) = (n>1) && !isprime(n) && (f(n) <= n); \\ Michel Marcus, Jul 17 2019