bibkey: ramanujan1915highlycomposite authors: Srinivasa Ramanujan year: 1915 title: Highly composite numbers doi: 10.1112/plms/s2_14.1.347 url: https://ramanujan.sirinudi.org/Volumes/published/ram15.html claim: Superior highly composite numbers optimize the divisor count divided by a positive power of the integer; prime exponent thresholds and the 5040 divisor record are classical inputs. strata_touched: [] license: citation-only triage: anchor

Highly composite numbers

Proceedings of the London Mathematical Society, second series 14, 347–409. The primary text uses $d(N)$ for the positive divisor count, denoted $\tau (N)$ here.

Section 32, equations (182)–(183), defines a superior highly composite integer using $d(N)/N^{ϵ}$, with $ϵ>0$ and the stated convention for comparisons at tied prices. Sections 33–34, equations (185)–(197), give the prime exponent thresholds and the equivalent stack-of-primes description. A strictly interior price gives a unique maximizer.

Table 1 lists $5040=2^4{3}^{2}57$ with $d(5040)=60$ and marks it as superior highly composite. It also supplies the corresponding divisor record. The application at $ϵ=3/10$ uses exact adjacent integer thresholds; it is an application of this classical theory, not a new extremal theorem.

This objective differs from the colossally abundant objective $\sigma (n)/n^{1+ϵ}$. Neither the divisor-count price nor the record supplies a uniform Robin inequality or a physical cost law.