for all integers,
where
is the divisor function counting the positive
divisors of .
All superior highly composite numbers are highly
composite. Ramanujan (1962, p. 114) asserted that the quotient of consecutive
superior highly composite numbers is a prime number.
His argument has an unproved transcendence step (Waldschmidt 1992, pp. 1-7 and
1-8). The four exponentials conjecture
would imply the assertion.
The first few superior highly composite numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, ... (OEIS A002201), and the successive
multipliers, starting from 1, are the primes 2, 3,
2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, 7, 29, 3, 31, 2, 37, 41, 43, ...
(OEIS A000705).
Ramanujan, S. "Highly Composite Numbers." Proc. London Math. Soc.14, 347-407, 1915. Reprinted in Collected Papers
(Ed. G. H. Hardy et al. ). New York: Chelsea, pp. 78-129, 1962.Sloane,
N. J. A. Sequences A000705/M0423
and A002201/M1591 in "The On-Line Encyclopedia
of Integer Sequences."Waldschmidt, M. Linear Independence of
Logarithms of Algebraic Numbers. Madras, India: Institute of Mathematical Sciences,
1992. Version of Aug. 29, 2016. https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/LIL.pdf.
for which there is a real
number 
such that
,
where 
is the divisor function counting the positive
divisors of 
.
All superior highly composite numbers are highly
composite. Ramanujan (1962, p. 114) asserted that the quotient of consecutive
superior highly composite numbers is a prime number.
His argument has an unproved transcendence step (Waldschmidt 1992, pp. 1-7 and
1-8). The four exponentials conjecture
would imply the assertion.
