Question: Is $\log\omega$ an omnific integer? Is $\log\omega\in\mathbf{Oz}$?
Conway [ONAG, pg.46] defines an omnific integer to be divisible iff $x$ is divisible by every finite nonzero integer. Gonshor [Gon86, pg.156] defines a surreal number $x$ to be purely infinite if all the exponents in its normal form are positive. Thus, $x$ does not have a finite component and is a divisible number in $\mathbf{Oz}$.
Gonshor [pg.143] states that $\exp x$ can be defined in a uniform way for all surreal numbers $x$ and has the properties expected of an exponential function. He writes [pg.158] that $\exp\omega = \omega^\omega$. This illustrates a regrettable choice of notation: $\omega^\omega$ defined by the omega-map, should not be regarded as exponentiation. On [pg.161] $\log x$ is defined for all $x$ of the form $\omega^b$. Therefore, $\exp\omega = \omega^\omega$ implies $\omega = \log\omega^\omega$, but we cannot write $\log\omega^\omega = \omega\log\omega$.
Theorem 10.8[B] on page 164 states that ``$\log\omega^b$ is purely infinite for all $b$''. (Warning: there are two theorems labelled 10.8). With $b=1$, this implies $\log\omega \in\mathbf{Oz}$.
We therefore have, in [Gon96], an affirmative answer to the question posed, and may deduce: Answer: $\log\omega$ is an omnific integer?
However, this conclusion has some unexpected consequences. One example should suffice: if $\log\omega\in\mathbf{Oz}$ then, for all $b>0$, $\log_b\omega\in\mathbf{Oz}$. In particular, $\mu_2 = \log_2\omega\in\mathbf{Oz}$ and $\mu_3 = \log_3\omega\in\mathbf{Oz}$, Thus, $\omega = 2^{\mu_2}$ and $\omega = 3^{\mu_3}$. We remark that such bizarre factorization is not found for the hypernaturals [Gol98, Sec5.11, 3(c), pg.57]. Given this, and other strange results, the validity of Gonshor's result would appear to be in doubt.
References
[onag] Conway, John Horton, 1976: On Numbers and Games. Academic Press, London. 2nd Edn., 2001: CRC Press, Taylor and Francis Group. 242pp. ISBN: 978-1-568-81127-7.
[Gol98] Goldblatt, Robert, 1998: Lectures on the Hyperreals: an Introduction to Nonstandard Analysis. Graduate Texts in Mathematics (GTM), Vol.188. Springer-Verlag. 289pp. ISBN 978-0-3879-8464-3.
[Gon86] Gonshor, Harry, 1986: An Introduction to the Theory of Surreal Numbers. London Mathematical Society Lecture Note Series 110. Cambridge Univ.~Press, 161pp. ISBN: 9-780-5213-1205-9.
