%I #9 Feb 27 2024 07:37:00

%S 1,2,3,4,6,8,10,14,18,26

%N Integers k for which there is a lacunary modular form of weight k/2 which is a product of eta functions.

%C Borcherds remarks that this is also the list of numbers k such that there are modular forms on the orthogonal group O_{k,2}(R) which can be written as an "interesting infinite product".

%D R. E. Borcherds, (1994). Sporadic groups and string theory. In First European Congress of Mathematics: Paris, July 6-10, 1992 Volume I Invited Lectures (Part 1) (pp. 411-421). Basel: Birkhäuser Basel. [This is different from the article in the link below. Do not delete this reference.]

%H R. E. Borcherds, <a href="https://math.berkeley.edu/~reb/papers/ecm/ecm.pdf">Sporadic groups and string theory</a> (Expanded version of talk referenced above).

%H F. Dyson, <a href="https://doi.org/10.1090/S0002-9904-1972-12971-9">Missed opportunities</a>, Bull. Amer. Math. Soc. 78 (1972), 635-652.

%H J.-P. Serre, <a href="https://doi.org/10.1017/S0017089500006194">Sur la lacunarité des puissances de eta</a>, Glasgow Mathematical Journal 27 (1985) 203-221. [Borcherds remarks that this reference omits the number 18, however the form eta(q)^9*eta(q^2)^9 of weight 18/2 appears to be lacunary.]

%K nonn,fini,full

%O 1,2

%A _N. J. A. Sloane_, Feb 26 2024