APPENDIX: On Composite Numbers
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APPENDIX: On Composite Numbers sy J° L. Nicolas Highly composite numbers n are posi t ive integers sat isfying d(n) ~ d(m) for all m < n, (I) where d is the divisor function. Sr in ivasa Ramanujan studied h ighly composite numbers in great detai l and his long paper [3] is quite famous. But there was much work on highly composite numbers and related topics that Ramanujan did not publish. During his centennial in December 1987, the first publ ished copy [2] of his Lost Notebook and other unpubl ished papers was released and in this impressive volume a manuscr ipt of Ramanujan on h igh ly composite numbers (previously unpublished) is included (pages 280-308). It is to be noted, however, that at the top of page 295 of [2] the words - Middle of another paper is not handwri t ten by Ramanujan. A short analysis of this manuscr ipt on highly composite numbers is given in [i] po 238- 239. The table on page 280 of [2] is not a list of h ighly composite numbers. This table almost coincides with the list of largely composite numbers n which satisfy the weaker inequal ity d(n) > d(m) for all m ~ n.
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