Highly composite numbers

The name comes from Srinivasa Ramanujan, who devoted a long paper in the Proceedings of the London Mathematical Society in 1915 to numbers he called highly composite — those with more divisors than any smaller number. He had arrived at Trinity College, Cambridge, the previous year at G. H. Hardy's invitation, and the paper is one of the first substantial pieces of work he published from England. In it he proved the structure theorem that still underpins every efficient search: the prime factorisation of a highly composite number uses consecutive primes starting at 2, and the exponents never increase as the primes get larger. He also introduced a sparser subfamily, the superior highly composite numbers, as a tool for pinning down how fast the divisor count can grow.

The published paper was not the whole manuscript. Wartime paper shortages forced the Proceedings to cut it, and the remaining sections sat unpublished for decades until Jean-Louis Nicolas and Guy Robin edited and annotated them for The Ramanujan Journal in 1997.

Interest in such numbers long predates the terminology. In Book V of the Laws, Plato proposes 5040 as the number of landholders in his ideal city, specifically because of how many ways it divides — and 5040 is indeed highly composite, with 60 divisors. Whether Plato grasped the record-setting property or simply liked a convenient number is not settled; the mathematician Jean-Pierre Kahane suggested the former, but it remains a conjecture about Plato's intent rather than a documented claim.

The modern asymptotic picture begins with Paul Erdős, who showed in 1944 that the count of highly composite numbers below x grows at least as fast as a power of log x strictly greater than one. Nicolas and Robin extended that line of work through the 1970s and 1980s.

Key properties

• n is highly composite when d(n) > d(m) for every m < n. The sequence begins 1, 2, 4, 6, 12, 24, 36, 48, 60, 120.
• Every term greater than 1 factors over consecutive primes starting at 2, with non-increasing exponents: 2^a₁ · 3^a₂ · … · p^aₖ where a₁ ≥ a₂ ≥ … ≥ aₖ ≥ 1.
• That final exponent aₖ equals 1 for every highly composite number except two: 4 = 2² and 36 = 2²·3².
• 1 is the only odd term, and 1, 4 and 36 are the only perfect squares in the whole sequence.
• Every term greater than 6 is abundant — its divisors excluding itself add up to more than the number.
• 720720 is the smallest number with 240 divisors, and nothing below one million has more.
• The 136th term, 10,108,248,702,552,000, is the first to exceed 2⁵³−1, so this page computes with arbitrary-precision integers.
• There are infinitely many, since d(n) is unbounded; Erdős proved in 1944 that the number of them below x exceeds (log x)^(1+c) for some c > 0.

Where they turn up

• Plato’s Laws, Book V, sets the ideal city at 5040 households and remarks on how many ways that splits — 5040 has 60 divisors, so 59 besides itself.
• The familiar divisions of time and angle — 12, 24, 60 and 360 — are all highly composite, though they descend from Babylonian sexagesimal counting rather than from anyone deliberately maximising divisors.
• Retail and packaging counts cluster on 12, 24, 48, 60 and 120 precisely because they split evenly so many ways; this is a practical convention, not a mathematical result.
• Ramanujan’s superior highly composite numbers are the tool used to establish the maximal order of the divisor function.
• The number 5040 is the exact threshold in Robin’s 1984 criterion, which makes the Riemann hypothesis equivalent to a bound on divisor sums for all n above it.
• Programmers sometimes call these "anti-primes" in puzzle and interview contexts — an informal nickname rather than standard terminology.