A natural number n is called highly composite, if it has more divisors than any smaller number. Let the prime-factorization of such a number n be p1e1p2e2p3e3...with pj > pi if j > i and the exponents ei non-negative integers. Generally the number of divisors of n is then (e1+1)(e2+1)(e3+1)... and we have for highly composite numbers: if j > i then ej <= ei. Lastly, denote by m the largest index i such that ei > 0, also called the maximal prime index. The average prime exponent ei of such a hihly composite number is proportional to 1 / ln pi. Here is a list of the first 1200 highly composite numbers computed 1991-10-31 on a HP 710/50 MHz.
Recently I discovered an Algorithm for Generating Highly Composite Numbers
and I recognized my misspelling "composited" :-/.
Therefore I dug up my old C-source for comparision.
The algorithm uses as non-triviality only S. Ramanujan's formula [ln(p)/ln(q)] <= aq < = 2 [ln(p+)/ln(q)] to estimate the prime-exponents aq of any prime q by the highest occuring prime p and its successor prime p+.
I think the program needed about 4 minutes to compute the smallest 1200 HCNs on the HP 710. Anyway, on an old K6-2 233 MHz it needs 49 sec for the first 1000 HCNs and
on an modern Athlon Thunderbird 1,2 GHz it needs < 6.5 sec.
I think this remarkable speed advantage is due to the fact that it is a C-program not a Mathematica-code.
But the real surprise was, that in the above cited PDF the 1000th HCN is given as 28365473112...157 163 which is my 1001-th ! Because for the 97-th HCN we got the same, it
seems that there is a HCN missing from Donald Siano's program!.
See his Web-site.
After an email exchange --- and the discovery of the miss of the 769-th HCN in his list ---
he said this publication/algorithm was only "a joke", "an entertainment",
not serious mathematics --- dispite he let check his result from a number theorist (Kiran Kedlaya) and then stated its correctness
(the claimed 1000-th HCN of his HCN-list is
indeed the 1000-th HCN) by an unpublished different algorithm.
Rerunning 2002-11-08 my C-program up to the 10000-th HCN enables this diagram:
Recall the definition of a superior highly composite number, SHCN, which
implies ei = [1/(pi1/x-1)] for a certain x > 0.
Considering this sub-class of the HCNs, we
see that the asymptotics of the highest prime index m is given by an estimation of Pi(ln(SHCN)):
Anyway, bounds on m results in a remarkable speed-up of the respectively modified algorithm: e.g. the first 10000 HCNs in less than half a minute compared to half an hour without these heuristics!
How many HCNs are there up to a given bound x?
Erdös proved: asymptotically at least ln1+eps(x).
From the following diagram we get the impression that k, the index of the k-th HCN, is perhaps proportional to ln(HCN)1.25.
But here you see the long term variations on k better:
These made the estimation of the true exponent numerically very difficult.
And lastly have a look how d(HCN) is approximated by ln(HCN):
Next a gziped list of the proved first 124260 HCNs (1.75 MB) ---
this was done by a new implemented algorithm using interval arithmetic in december 2002 - january 2003.
Each line of this uncompressed file contains a HCN in the format:
ln(HCN) ln(d(HCN)) m e_1 e_2 e_3 ... e_i^k_i ... 2^k_2 1^k1.
1st entry: ln(HCN) the natural logarithm of the HCN up to 6 digits.
2nd entry: ln(d(HCN)) the natural logarithm of the divisor-function up to 7 digits
3rd entry: m the number of different primes in the factorization of the HCN
remaining entries: e_1 e_2 ,,,, the list of the exponents of its canonical prime-factorization.
Here successive equal exponents are coded as e_i^k_i, meaning k_i identical
with exponent value e_i follow.
Finally a working-paper on the distribution of HCNs as DVI-File. The last two conjectures of this paper are used as bounds on m.
With this heuristic I computed the list of the proven smallest 779674 HCNs (1.5 MB).
Due to save space, each line of this 'unbzip2'ed file misses the ln(HCN) and ln(d(HCN)) entry and so starts with
the 3rd entry in the format described above for the first 1039000 HCNs.
I forgot to publish the (now probably outdated) final version of my above mentioned working-paper on the distribution of HCNs from August 2012 as a PS-file, because it contains two encapsulated-PS-grafics.
Early November 2025 I stumpled about the paper
Highly Composite Numbers and the Riemann hypothesis by Jean-Louis Nicolas which appeared in Ramanujan Journal V. 57 (2022).
The two sentences "Consequently, formulas (3.28) and (3.29) yield an algorithm to compute all integers n such that benε(n) ≤ B for a B not too large.
With B close to ε/3, this algorithm has been used in [26] to compute the hc numbers between two consecutive
shc numbers of common parameter ε." in front of lemma 3.16 in the section 3.5 Benefit triggered my very interest to design and write a new algorithm to compute HCNs based on this function benε(n).
From this algorithm a C-program (compiled by GCC 15.2.1) emerged which was able to compute all HCNs ≤ 108 until 1st December in less than 48 CPU-hours using a Xeon E3-1275v5 @ 3.7 GHz.
Here you can see the maximal value of
benε(HCN)/ε = ln(HCN/SHCN)-(ln(d(HCN)/d(SHCN)))/ε for the smallest 400000 SHCNs(ε):
For download I offer the xz-compressed list of the first 100181452 HCNs of 44 MB which expands to 3.7 GB decompressed.
Then each line represents a HCN n which is given
by its list of non-zero prime number exponents with multiplicities of the same exponent-value are counted by ^.
A lead-in of a line by s indicates it is also a SHCN. Due to save further space the exponent-value is omitted
until line end as long as the previous exponent-value is 1 larger than the current value and all further exponent-values are decreasing by exactly 1 down to the last exponent value of 1. The maximal number of different exponents for every HCN in this data set is 12 and it needs at most the 177621 first prime numbers.
Moreover I like to mention On highly composite numbers by Jean-Louis Nicolas which appeared in "Ramanujan Revisited", Proc. Centenary Conference, Univ. Illinois, Urbana, Academic Press, NY (1987), p. 215-244.
Here he gives some estimations of the asymptotic growth of the number of HCNs less than or equal a given positive number x, which is denoted by Q(x).
I like to cite the conjecture
limx → ∞ ln(Q(x)) / ln ln(x) = ln(30)/ln(16) = 1.2267226489...
by Jean-Louis Nicolas 1987
and his proved inequality
liminfx → ∞ ln(Q(x)) / ln ln(x) ≥ 1 + (1-τ)ln(15/8)/ln(8) ≥ 1.143591...
with τ <= 0.525 proved by Baker, Harman and Pintz in 2001.
BTW: Jean-Louis Nicolas is a French number theorist born 1942
and well-known expert regarding HCNs.
no HCN=n ln(n) d(n) ln(d(n)) SHCN Ome factorization k exponents shorter ------------------------------------------------------------------------------------------------------------------ 1 1 0.000000000000 1 0.000000000000 s 0 0 2 2 0.693147180560 2 0.693147180560 s 1 2 1 1 1 3 4 1.386294361120 3 1.098612288668 1 2^2 1 2 2 4 6 1.791759469228 4 1.386294361120 s 2 2 3 1 1^2 1^2 5 12 2.484906649788 6 1.791759469228 s 2 2^2 3 2 2 1 2^1^1 6 24 3.178053830348 8 2.079441541680 2 2^3 3 2 3 1 3 1 7 36 3.583518938456 9 2.197224577336 2 2^2 3^2 1 2^2 2^2 8 48 3.871201010908 10 2.302585092994 2 2^4 3 2 4 1 4 1 9 60 4.094344562222 12 2.484906649788 s 3 2^2 3 5 2 2 1^2 2^1^2 10 120 4.787491742782 16 2.772588722240 s 3 2^3 3 5 2 3 1^2 3 1^2 11 180 5.192956850890 18 2.890371757896 3 2^2 3^2 5 2 2^2 1 2^2^1 12 240 5.480638923342 20 2.995732273554 3 2^4 3 5 2 4 1^2 4 1^2 13 360 5.886104031450 24 3.178053830348 s 3 2^3 3^2 5 3 3 2 1 3^1^1^1 14 720 6.579251212010 30 3.401197381662 3 2^4 3^2 5 3 4 2 1 4 2^1^1 15 840 6.733401891837 32 3.465735902800 4 2^3 3 5 7 2 3 1^3 3 1^3 16 1260 7.138866999946 36 3.583518938456 4 2^2 3^2 5 7 2 2^2 1^2 2^2^2 17 1680 7.426549072397 40 3.688879454114 4 2^4 3 5 7 2 4 1^3 4 1^3 18 2520 7.832014180505 48 3.871201010908 s 4 2^3 3^2 5 7 3 3 2 1^2 3^1^1^2 19 5040 8.525161361065 60 4.094344562222 s 4 2^4 3^2 5 7 3 4 2 1^2 4 2^1^2 20 7560 8.930626469174 64 4.158883083360 4 2^3 3^3 5 7 2 3^2 1^2 3^2 1^2 21 10080 9.218308541625 72 4.276666119016 4 2^5 3^2 5 7 3 5 2 1^2 5 2^1^2 22 15120 9.623773649734 80 4.382026634674 4 2^4 3^3 5 7 3 4 3 1^2 4 3 1^2 23 20160 9.911455722185 84 4.430816798843 4 2^6 3^2 5 7 3 6 2 1^2 6 2^1^2 24 25200 10.134599273500 90 4.499809670330 4 2^4 3^2 5^2 7 3 4 2^2 1 4 2^2^1 25 27720 10.229909453304 96 4.564348191468 5 2^3 3^2 5 7 11 3 3 2 1^3 3^1^1^3 26 45360 10.722385938402 100 4.605170185988 4 2^4 3^4 5 7 2 4^2 1^2 4^2 1^2 27 50400 10.827746454059 108 4.682131227124 4 2^5 3^2 5^2 7 3 5 2^2 1 5 2^2^1 28 55440 10.923056633864 120 4.787491742782 s 5 2^4 3^2 5 7 11 3 4 2 1^3 4 2^1^3 29 83160 11.328521741972 128 4.852030263920 5 2^3 3^3 5 7 11 2 3^2 1^3 3^2 1^3 30 110880 11.616203814424 144 4.969813299576 5 2^5 3^2 5 7 11 3 5 2 1^3 5 2^1^3 31 166320 12.021668922532 160 5.075173815234 5 2^4 3^3 5 7 11 3 4 3 1^3 4 3 1^3 32 221760 12.309350994984 168 5.123963979403 5 2^6 3^2 5 7 11 3 6 2 1^3 6 2^1^3 33 277200 12.532494546298 180 5.192956850890 5 2^4 3^2 5^2 7 11 3 4 2^2 1^2 4 2^2^2 34 332640 12.714816103092 192 5.257495372028 5 2^5 3^3 5 7 11 3 5 3 1^3 5 3 1^3 35 498960 13.120281211200 200 5.298317366548 5 2^4 3^4 5 7 11 2 4^2 1^3 4^2 1^3 36 554400 13.225641726858 216 5.375278407684 5 2^5 3^2 5^2 7 11 3 5 2^2 1^2 5 2^2^2 37 665280 13.407963283652 224 5.411646051855 5 2^6 3^3 5 7 11 3 6 3 1^3 6 3 1^3 38 720720 13.488005991325 240 5.480638923342 s 6 2^4 3^2 5 7 11 13 3 4 2 1^4 4 2^1^4 39 1081080 13.893471099433 256 5.545177444480 6 2^3 3^3 5 7 11 13 2 3^2 1^4 3^2 1^4 40 1441440 14.181153171885 288 5.662960480136 s 6 2^5 3^2 5 7 11 13 3 5 2 1^4 5 2^1^4 41 2162160 14.586618279993 320 5.768320995794 6 2^4 3^3 5 7 11 13 3 4 3 1^4 4 3 1^4 42 2882880 14.874300352445 336 5.817111159963 6 2^6 3^2 5 7 11 13 3 6 2 1^4 6 2^1^4 43 3603600 15.097443903759 360 5.886104031450 6 2^4 3^2 5^2 7 11 13 3 4 2^2 1^3 4 2^2^3 44 4324320 15.279765460553 384 5.950642552588 s 6 2^5 3^3 5 7 11 13 3 5 3 1^4 5 3 1^4 45 6486480 15.685230568662 400 5.991464547108 6 2^4 3^4 5 7 11 13 2 4^2 1^4 4^2 1^4 46 7207200 15.790591084319 432 6.068425588244 6 2^5 3^2 5^2 7 11 13 3 5 2^2 1^3 5 2^2^3 47 8648640 15.972912641113 448 6.104793232415 6 2^6 3^3 5 7 11 13 3 6 3 1^4 6 3 1^4 48 10810800 16.196056192428 480 6.173786103902 6 2^4 3^3 5^2 7 11 13 4 4 3 2 1^3 4^1^1^1^3 49 14414400 16.483738264879 504 6.222576268071 6 2^6 3^2 5^2 7 11 13 3 6 2^2 1^3 6 2^2^3 50 17297280 16.666059821673 512 6.238324625040 6 2^7 3^3 5 7 11 13 3 7 3 1^4 7 3 1^4 51 21621600 16.889203372987 576 6.356107660696 s 6 2^5 3^3 5^2 7 11 13 4 5 3 2 1^3 5 3^1^1^3 52 32432400 17.294668481096 600 6.396929655216 6 2^4 3^4 5^2 7 11 13 3 4^2 2 1^3 4^2 2^1^3 53 36756720 17.419831624050 640 6.461468176354 7 2^4 3^3 5 7 11 13 17 3 4 3 1^5 4 3 1^5 54 43243200 17.582350553547 672 6.510258340523 6 2^6 3^3 5^2 7 11 13 4 6 3 2 1^3 6 3^1^1^3 55 61261200 17.930657247816 720 6.579251212010 7 2^4 3^2 5^2 7 11 13 17 3 4 2^2 1^4 4 2^2^4 56 73513440 18.112978804610 768 6.643789733148 7 2^5 3^3 5 7 11 13 17 3 5 3 1^5 5 3 1^5 57 110270160 18.518443912718 800 6.684611727668 7 2^4 3^4 5 7 11 13 17 2 4^2 1^5 4^2 1^5 58 122522400 18.623804428376 864 6.761572768804 7 2^5 3^2 5^2 7 11 13 17 3 5 2^2 1^4 5 2^2^4 59 147026880 18.806125985170 896 6.797940412975 7 2^6 3^3 5 7 11 13 17 3 6 3 1^5 6 3 1^5 60 183783600 19.029269536484 960 6.866933284462 7 2^4 3^3 5^2 7 11 13 17 4 4 3 2 1^4 4^1^1^1^4freely accessable original paper of S. Ramanujan published 1915