22
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When dinosaurs ruled the earth, one of my assignments in a Problem Seminar in undergraduate was to devise and prove a conjecture about antiprimes, and this was my attempt:

An antiprime (also called a highly composite number) is a positive integer that has more divisors than any number less than it. The first few antiprimes are $$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360,...$$ Conjecture: For every antiprime $n>1$, there is a prime $p$ such that $p\mid n$ and $n/p$ is an antiprime.

Anyways, I never found a proof for that, and about ten years ago I asked the xkcd math forums if they could guide me. Instead, someone posted a counterexample that was quite enormous.

A question on MESE is asking about relatively elementary mathematical conjectures whose smallest counterexamples are large numbers. I'd like to suggest my problem, but the xkcd forums went down five months ago over a data breach and my thread wasn't cached by Google or the Wayback Machine.

Can someone find that counterexample? The person who posted didn't indicate if they came up with their number through mathematical reasoning or programming. I discovered this morning that OEIS has a list of the first ten thousand antiprimes, so in theory it may just come down to finding the prime decomposition of each of them. But, if possible, is there a mathematical argument that would lead one to the correct number?

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    $\begingroup$ What a challenging problem! (this one computer once said). $\endgroup$ – Piquito Jan 31 '20 at 12:44
  • $\begingroup$ downloaded the first 779674 HCN by Flammenkamp. wwwhomes.uni-bielefeld.de/achim/highly.html $\endgroup$ – Will Jagy Feb 2 '20 at 16:39
  • $\begingroup$ After a few days of programming, I have posted the first 35 such numbers. The first three numbers on the line are the total exponent, the number of distinct primes, and the number of "pairs". Each pair reads ( m,n) which means the next n consecutive primes each has the exponent m. $\endgroup$ – Will Jagy Feb 5 '20 at 20:33
11
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My original answer was different since it was based on flawed code snippet for generating these numbers. However I've looked at the list that you have found as well, parsed it and found that smallest counterexample to your conjecture is $$362279431624673937974303738230488502933082643722886373107941760000$$ which is the $815$th highly composite number. To check quickly from the given list, no prime decomposition is necessary. All we need to do is check that none of the $n/d$ is prime for all antiprimes $d<n$. Otherwise, we could take $p=n/d$ and it would satisfy the condition of conjecture. Same works in the opposite direction: if there would be a prime $p$, take $d=n/p$, and since clearly $d<n$, we have $n/d=p$ is a prime.

Unfortunately, I do not know if there is a mathematical reasoning alone that could help you to get to that number. The above is just assuming we already have a list.

Here is a Python snippet I used:

import sympy

L=set()
for line in open("b002182.txt").readlines():
    n = int(line.split()[1])
    isok = False
    for prev in L:
        if n % prev == 0:
            if sympy.isprime(n // prev):
                isok = True
                break
    if not isok and n > 1:
        print(line)
        break

    L.add(n)

Here is also list of (smallest) witness primes $p$ for all the preceeding numbers, $n/p$ and also prime divisors: https://gist.github.com/TheSil/f26dc0a516d12a9a556ada3191512c99

Check also An Algorithm for Computing Highly Composite Numbers article, and on the site related post is there a large list of highly composite number?.

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    $\begingroup$ I will say that this is larger than I recall the answer I got ten years ago was, but they may have done the same thing you did on your first draft. Still, phew! As I said ten years ago, this is really close to infinity! $\endgroup$ – Matthew Daly Jan 31 '20 at 14:22
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    $\begingroup$ I have double checked your counterexample. Since the largest prime factor is only $137$ we can simply work out a few dozen quotients and see if any are in the list. Seeing none, I give +1. $\endgroup$ – Oscar Lanzi Jan 31 '20 at 19:06
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    $\begingroup$ @OscarLanzi Thank you for checking. I have now added also file that contains list of witness primes for all the preceeding numbers, as well as prime divisors. $\endgroup$ – Sil Jan 31 '20 at 19:53
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    $\begingroup$ @Sil I'm learning a lot today about how much Google Search can know about a webpage that it claims isn't cached! $\endgroup$ – Matthew Daly Jan 31 '20 at 20:33
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    $\begingroup$ @MatthewDaly suggest you both write correct programs for Ramanujan's Superior Highly Composite numbers; I discuss those at one of the linked questions. Thanks for a fun question. For that matter, I had not known of Kedlaya's new algorithm. $\endgroup$ – Will Jagy Jan 31 '20 at 20:34
5
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ADDED, evening. I am running the program on Flammenkamp's entire dataset. First I put it in reverse order, based on the first number in each line, which is the sum of the exponents of all the primes. Once again, there are two neighboring examples that are very similar. The third and fourth lines below both indicate 3831 distinct primes and exactly 3740 of those primes with exponent 1. Lots of such pairs in the first 35 examples.

We found 5 of these numbers before. The sixth one is about $2.58697067953 \cdot 10^{857}\; , \;$ with largest prime factor $1907.$ The seventh one is about $1.1815511968 \cdot 10^{947}\; , \;$ with largest prime factor $2113.$ The eighth one is about $1.701433723433 \cdot 10^{948}\; , \;$ with repeat of largest prime factor $2113.$ The ninth one is about $3.90407489941 \cdot 10^{968}\; , \;$ with largest prime factor $2153.$

The 76th example is about $1.4343006428558 \cdot 10^{\color{red}{16866}} \; , \; $ with largest prime factor $ 38501 \; . \; $ It begins

$$2^{19} 3^{14} 5^8 7^7 11^5 13^5 17^4 19^4 23^4 29^4 \cdots 38459 \cdot 38461 \cdot 38501$$

After playing with this for several days, I now see little reason that there should be just a finite set of these examples. The rule, the crucial rule, is that any highly composite number is a product of primorials: the prime factorization is consecutive primes from $2$ up to some prime, with the requirement that the exponents of these primes be non-increasing. What this means is that there are a very small number of primes by which our number $n$ might be divided, and the count of these is the third number in each row of my printout. Tiny. This is also the number of $(m,n)$ pairs in that line.

Back to the sixth example:

 = 2^12 3^9 5^6 7^5 11^4 13^3 17^3 19^3 23^3 29^2 31^2 37^2 41^2 43^2 47^2 53^2 59^2 61^2 67^2 71^2 73^2 79^2 83^2 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877 1879 1889 1901  1907

 log ten  857.413

Below are the first 76 highly composite numbers that are not a prime times a highly composite number. The notation is that of Flammenkamp, the list where he says

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

enter image description here


the first 76 examples:

   55   33  5: ( 10, 1)( 6, 1)( 4, 1)( 2, 5)( 1, 25)
  123   92  6: ( 10, 1)( 7, 1)( 5, 1)( 3, 3)( 2, 6)( 1, 80)
  132  100  6: ( 10, 1)( 7, 1)( 5, 1)( 3, 3)( 2, 7)( 1, 87)
  139  104  6: ( 12, 1)( 8, 1)( 4, 2)( 3, 1)( 2, 9)( 1, 90)
  141  106  6: ( 12, 1)( 8, 1)( 5, 1)( 3, 2)( 2, 9)( 1, 92)
  345  292  8: ( 12, 1)( 9, 1)( 6, 1)( 5, 1)( 4, 1)( 3, 4)( 2, 14)( 1, 269)
  372  319  8: ( 13, 1)( 8, 1)( 6, 1)( 5, 1)( 4, 1)( 3, 3)( 2, 16)( 1, 295)
  376  319  7: ( 16, 1)( 10, 1)( 5, 2)( 4, 1)( 3, 3)( 2, 16)( 1, 295)
  379  325  8: ( 12, 1)( 9, 1)( 6, 1)( 5, 1)( 4, 1)( 3, 4)( 2, 15)( 1, 301)
  386  330  6: ( 15, 1)( 10, 1)( 5, 2)( 3, 5)( 2, 15)( 1, 306)
  415  357  7: ( 14, 1)( 10, 1)( 6, 1)( 4, 3)( 3, 3)( 2, 16)( 1, 332)
  420  363  8: ( 13, 1)( 9, 1)( 6, 1)( 5, 1)( 4, 1)( 3, 4)( 2, 17)( 1, 337)
  420  363  8: ( 14, 1)( 8, 1)( 6, 1)( 5, 1)( 4, 1)( 3, 4)( 2, 17)( 1, 337)
  456  394  7: ( 16, 1)( 10, 1)( 5, 2)( 4, 1)( 3, 4)( 2, 19)( 1, 366)
  467  407  7: ( 15, 1)( 10, 1)( 6, 1)( 4, 3)( 3, 3)( 2, 17)( 1, 381)
  467  407  7: ( 16, 1)( 9, 1)( 6, 1)( 4, 3)( 3, 3)( 2, 17)( 1, 381)
  483  421  8: ( 13, 1)( 10, 1)( 6, 1)( 5, 1)( 4, 2)( 3, 3)( 2, 20)( 1, 392)
  613  546  8: ( 14, 1)( 9, 1)( 7, 1)( 5, 1)( 4, 2)( 3, 5)( 2, 20)( 1, 515)
  625  556  8: ( 16, 1)( 10, 1)( 6, 1)( 5, 1)( 4, 2)( 3, 3)( 2, 24)( 1, 523)
  642  573  8: ( 15, 1)( 10, 1)( 7, 1)( 5, 1)( 4, 1)( 3, 6)( 2, 21)( 1, 541)
  670  598  8: ( 16, 1)( 10, 1)( 7, 1)( 5, 1)( 4, 1)( 3, 6)( 2, 23)( 1, 564)
  677  607  8: ( 16, 1)( 9, 1)( 6, 1)( 5, 1)( 4, 2)( 3, 4)( 2, 24)( 1, 573)
  712  641  8: ( 15, 1)( 9, 1)( 7, 1)( 5, 1)( 4, 2)( 3, 5)( 2, 23)( 1, 607)
  721  647  8: ( 16, 1)( 10, 1)( 7, 1)( 5, 1)( 4, 2)( 3, 4)( 2, 26)( 1, 611)
  722  647  8: ( 18, 1)( 10, 1)( 6, 1)( 5, 1)( 4, 2)( 3, 4)( 2, 26)( 1, 611)
  852  774  8: ( 17, 1)( 9, 1)( 7, 1)( 5, 1)( 4, 3)( 3, 4)( 2, 27)( 1, 736)
  937  857  8: ( 16, 1)( 10, 1)( 7, 1)( 5, 1)( 4, 2)( 3, 5)( 2, 30)( 1, 816)
  939  856  7: ( 18, 1)( 10, 1)( 6, 2)( 4, 3)( 3, 4)( 2, 30)( 1, 815)
  950  869  7: ( 16, 1)( 10, 1)( 6, 2)( 4, 3)( 3, 5)( 2, 28)( 1, 829)
 1117 1027  9: ( 17, 1)( 10, 1)( 7, 1)( 6, 1)( 5, 1)( 4, 2)( 3, 5)( 2, 34)( 1, 981)
 1138 1049  8: ( 17, 1)( 10, 1)( 7, 1)( 5, 1)( 4, 3)( 3, 6)( 2, 33)( 1, 1003)
 1140 1051  8: ( 16, 1)( 11, 1)( 7, 1)( 5, 1)( 4, 3)( 3, 6)( 2, 33)( 1, 1005)
 1149 1059  9: ( 16, 1)( 10, 1)( 7, 1)( 6, 1)( 5, 1)( 4, 2)( 3, 6)( 2, 33)( 1, 1013)
 1161 1069  8: ( 19, 1)( 11, 1)( 7, 1)( 5, 1)( 4, 4)( 3, 4)( 2, 34)( 1, 1023)
 1239 1147  8: ( 17, 1)( 10, 1)( 7, 1)( 6, 1)( 4, 4)( 3, 5)( 2, 34)( 1, 1100)
 1424 1325  8: ( 19, 1)( 12, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 7)( 2, 33)( 1, 1277)
 1444 1344  8: ( 19, 1)( 12, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 7)( 2, 34)( 1, 1295)
 1447 1349  8: ( 18, 1)( 11, 1)( 7, 1)( 5, 1)( 4, 4)( 3, 7)( 2, 35)( 1, 1299)
 1478 1378  8: ( 18, 1)( 11, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 7)( 2, 36)( 1, 1327)
 1480 1378  8: ( 19, 1)( 12, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 7)( 2, 36)( 1, 1327)
 1540 1438  9: ( 16, 1)( 11, 1)( 8, 1)( 6, 1)( 5, 1)( 4, 3)( 3, 7)( 2, 38)( 1, 1385)
 1540 1438  9: ( 17, 1)( 10, 1)( 8, 1)( 6, 1)( 5, 1)( 4, 3)( 3, 7)( 2, 38)( 1, 1385)
 1581 1475  9: ( 18, 1)( 12, 1)( 8, 1)( 6, 1)( 5, 1)( 4, 3)( 3, 6)( 2, 41)( 1, 1420)
 1818 1708  9: ( 17, 1)( 12, 1)( 7, 1)( 6, 1)( 5, 2)( 4, 2)( 3, 8)( 2, 42)( 1, 1650)
 1870 1758  9: ( 18, 1)( 11, 1)( 8, 1)( 6, 1)( 5, 1)( 4, 3)( 3, 7)( 2, 46)( 1, 1697)
 1936 1822  9: ( 19, 1)( 10, 1)( 8, 1)( 6, 1)( 5, 1)( 4, 4)( 3, 7)( 2, 45)( 1, 1761)
 1938 1824  9: ( 19, 1)( 11, 1)( 8, 1)( 6, 1)( 5, 1)( 4, 3)( 3, 7)( 2, 47)( 1, 1762)
 2006 1891  9: ( 19, 1)( 10, 1)( 8, 1)( 6, 1)( 5, 1)( 4, 4)( 3, 7)( 2, 46)( 1, 1829)
 2329 2207  9: ( 18, 1)( 13, 1)( 7, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 9)( 2, 47)( 1, 2142)
 2367 2245  9: ( 19, 1)( 12, 1)( 7, 1)( 6, 1)( 5, 1)( 4, 4)( 3, 9)( 2, 48)( 1, 2179)
 2368 2245  9: ( 18, 1)( 13, 1)( 7, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 9)( 2, 48)( 1, 2179)
 2510 2385  9: ( 18, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 2)( 3, 10)( 2, 50)( 1, 2317)
 2547 2420  9: ( 18, 1)( 12, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 8)( 2, 54)( 1, 2349)
 2565 2437  9: ( 19, 1)( 11, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 8)( 2, 54)( 1, 2366)
 2611 2482  9: ( 18, 1)( 13, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 8)( 2, 55)( 1, 2410)
 2683 2555  9: ( 18, 1)( 11, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 9)( 2, 54)( 1, 2483)
 2856 2724  9: ( 19, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 1)( 4, 4)( 3, 9)( 2, 56)( 1, 2650)
 2961 2826  9: ( 20, 1)( 12, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 2)( 3, 11)( 2, 57)( 1, 2750)
 2989 2855  9: ( 20, 1)( 12, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 2)( 3, 10)( 2, 58)( 1, 2779)
 3029 2895  9: ( 18, 1)( 12, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 9)( 2, 59)( 1, 2818)
 3111 2973  9: ( 20, 1)( 11, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 9)( 2, 61)( 1, 2894)
 3144 3003  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 9)( 2, 62)( 1, 2923)
 3145 3003  9: ( 22, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 9)( 2, 62)( 1, 2923)
 3169 3029  9: ( 20, 1)( 13, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 9)( 2, 62)( 1, 2949)
 3183 3042  9: ( 22, 1)( 12, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 9)( 2, 62)( 1, 2962)
 3299 3159  9: ( 18, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 11)( 2, 60)( 1, 3079)
 3565 3415  9: ( 21, 1)( 13, 1)( 9, 1)( 6, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 63)( 1, 3331)
 3566 3417  9: ( 21, 1)( 13, 1)( 9, 1)( 6, 1)( 5, 2)( 4, 4)( 3, 10)( 2, 64)( 1, 3333)
 3587 3440  9: ( 20, 1)( 13, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 11)( 2, 65)( 1, 3355)
 3588 3440  9: ( 22, 1)( 12, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 11)( 2, 65)( 1, 3355)
 3688 3537  9: ( 22, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 11)( 2, 67)( 1, 3450)
 3702 3554  9: ( 22, 1)( 11, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 12)( 2, 63)( 1, 3470)
 3985 3831  9: ( 20, 1)( 13, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 3)( 3, 12)( 2, 70)( 1, 3740)
 3987 3831  9: ( 22, 1)( 12, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 70)( 1, 3740)
 4121 3964  9: ( 22, 1)( 12, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 71)( 1, 3872)
 4217 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966)

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I concur with Sil and Will's lowest such highly composite number as $$362279431624673937974303738230488502933082643722886373107941760000$$

Define $h_n$ as the $n$'th highly composite number. I also find that there are $5$ such examples less than $10^{600}$:

  • $h_{815}=362\,279\,431\ldots000\approx 10^{66}$

  • $h_{4372 }=604\,369\,999\ldots000\approx 10^{220}$

  • $h_{4996 }=111\,588\,694\ldots000\approx 10^{244}$

  • $h_{5312 }=605\,226\,260\ldots000\approx 10^{257}$

  • $h_{5442 }=142\,430\,057\ldots000\approx 10^{262}$

The full form of the numbers is available (here). I used a python code adapted from dario2994's Github: generate_hcn.py code on Github. The code in question is available at Github: DalyConjecture.py and generates the list of highly composite numbers and primes intrinsically, so requires no prerequisite data tables nor packages. It's quite a fast running code and returns the relevant highly composite numbers below $10^{200}$ within seconds but (on my machine at least) is susceptible to memory overflows when searching through high bounds $\approx 10^{700}$.

The observation that the counterexamples seem clustered in $(10^{66},10^{262})$ yet absent from the wide interval $(10^{262},10^{600})$ suggests that these might be the only counterexamples. I've not yet been able to check beyond $10^{600}$.


Following Will Jagy's comment, I've included the prime decompositions of the counterexamples:

  • $h_{815}=2^{10}\cdot3^{6}\cdot5^{4}\cdot7^{2}\cdot\ldots\cdot131^{1}\cdot \underbrace{137^{1}}_{p_{34}}$

  • $h_{4372 }=2^{10}\cdot3^{7}\cdot5^{5}\cdot7^{3}\cdot\ldots\cdot463^{1}\cdot \underbrace{467^{1}}_{p_{92}}$

  • $h_{4996 }=2^{10}\cdot3^{7}\cdot5^{5}\cdot7^{3}\cdot\ldots\cdot521^{1}\cdot \underbrace{523^{1}}_{p_{100}}$

  • $h_{5312 }=2^{12}\cdot3^{8}\cdot5^{4}\cdot7^{4}\cdot\ldots\cdot557^{1}\cdot \underbrace{563^{1}}_{p_{104}}$

  • $h_{5442 }=2^{12}\cdot3^{8}\cdot5^{5}\cdot7^{3}\cdot\ldots\cdot569^{1}\cdot \underbrace{571^{1}}_{p_{106}}$

We see that the counterexamples have monotonically decreasing prime exponents, so they are also superior highly composite numbers. It is interesting that the largest prime factor has even index, though the sequence of these indices has no obvious pattern.

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  • $\begingroup$ We actually have an example at log10 = 66, then another gap, tgen several between 200 and 300 before what seems to be another gap. Are the examples continuing to occur in clusters? $\endgroup$ – Oscar Lanzi Jan 31 '20 at 22:25
  • $\begingroup$ @OscarLanzi The largest upper limit I can use without an overflow is $10^{500}$ so I can't say. But it is certainly curious that there seems to be no counterexamples greater than $10^{300}$. $\endgroup$ – Jam Jan 31 '20 at 22:33
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    $\begingroup$ Here is the early survey by Nicolas that got me interested. It does explain the SHC numbers, mentions Robin's computer method for HC numbers in between. I don't know of an English language article that gives all detail on Robin's algorithm. math.univ-lyon1.fr/~nicolas/hcnrevisited.pdf $\endgroup$ – Will Jagy Feb 1 '20 at 21:08
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    $\begingroup$ math.univ-lyon1.fr/~nicolas/publications.html and math.univ-lyon1.fr/~nicolas $\endgroup$ – Will Jagy Feb 1 '20 at 21:13
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    $\begingroup$ After a few days of programming, I have posted the first 35 such numbers. The first three numbers on the line are the total exponent, the number of distinct primes, and the number of "pairs". Each pair reads ( m,n) which means the next n consecutive primes each has the exponent m. $\endgroup$ – Will Jagy Feb 5 '20 at 20:32
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I worked up a full accounting of the 76th example. First, the example and the legal ways to divide it by a single prime, in such a way as to result in another number with non-increasing exponents. On each line, I began with log base ten of the number, then log base ten of its count of divisors. The "original" has a total of exponents 4217, so each "derived" number has total of exponents 4216. The number of distinct primes is 4059 in the original , this usually stay the same unless the largest prime factor is dropped, leading to distinct 4058.

The second section is the highly composite numbers with total exponent 4216, just a portion ( there is a 30,000 character limit here) but consecutive, in order both as to the indicated number and its count of divisors. Then I put the nine "derived" numbers in proper order within that list, showing that each of the nine numbers is not highly composite. A crucial aspect of this is trusting Flammenkamp's list of the first 779,674 highly composite numbers.

=========================================

16866.157 1243.547 4217 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966) original

16865.856 1243.5247 4216 4059  9: ( 18, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966)  derive 1
16865.68 1243.517 4216 4059  9: ( 19, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966)  derive 2
16865.458 1243.4958 4216 4059  8: ( 19, 1)( 14, 1)( 7, 2)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966)  derive 3
16865.312 1243.489 4216 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966)  derive 4
16865.043 1243.4678 4216 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 1)( 4, 5)( 3, 11)( 2, 72)( 1, 3966)  derive 5
16864.694 1243.4501 4216 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 12)( 2, 72)( 1, 3966)  derive 6
16864.293 1243.4221 4216 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 10)( 2, 73)( 1, 3966)  derive 7
16863.469 1243.3709 4216 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 71)( 1, 3967)  derive 8
16861.571 1243.246 4216 4058  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3965)  derive 9

==================================

16860.827 1243.1996 4216 4058  9: ( 20, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 70)( 1, 3966)
16860.974 1243.2072 4216 4058 10: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 6, 1)( 5, 1)( 4, 3)( 3, 12)( 2, 72)( 1, 3965)
16861.003 1243.2106 4216 4058  9: ( 19, 1)( 13, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 70)( 1, 3966)
16861.169 1243.2208 4216 4058 10: ( 20, 1)( 12, 1)( 8, 1)( 7, 1)( 6, 1)( 5, 1)( 4, 4)( 3, 12)( 2, 70)( 1, 3966)
16861.173 1243.2214 4216 4059  9: ( 22, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 69)( 1, 3968)
16861.349 1243.2343 4216 4059  9: ( 21, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 69)( 1, 3968)
16861.395 1243.2372 4216 4058  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3965)
16861.554 1243.2454 4216 4059  9: ( 20, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 5)( 3, 11)( 2, 68)( 1, 3969)
D16861.571 1243.246 4216 4058  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3965)  derive 9
16861.571 1243.2478 4216 4059  9: ( 21, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 69)( 1, 3968)
16861.747 1243.2598 4216 4059  9: ( 20, 1)( 13, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 69)( 1, 3968)
16861.83 1243.2652 4216 4058  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 71)( 1, 3965)
16861.982 1243.2734 4216 4059  9: ( 20, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 13)( 2, 67)( 1, 3969)
16862.278 1243.2932 4216 4059  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 3)( 4, 2)( 3, 13)( 2, 69)( 1, 3968)
16862.334 1243.299 4216 4059  9: ( 21, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 69)( 1, 3968)
16862.51 1243.311 4216 4059  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 69)( 1, 3968)
16862.732 1243.3245 4216 4059  9: ( 20, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 69)( 1, 3968)
16862.908 1243.3355 4216 4059  9: ( 19, 1)( 13, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 69)( 1, 3968)
16863.075 1243.3457 4216 4059 10: ( 20, 1)( 12, 1)( 8, 1)( 7, 1)( 6, 1)( 5, 1)( 4, 4)( 3, 12)( 2, 69)( 1, 3968)
16863.266 1243.3592 4216 4060  9: ( 21, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 68)( 1, 3970)
D16863.469 1243.3709 4216 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 71)( 1, 3967)  derive 8
16863.487 1243.3728 4216 4060  9: ( 21, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 68)( 1, 3970)
16863.552 1243.3782 4216 4059  9: ( 21, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 70)( 1, 3967)
16863.664 1243.3848 4216 4060  9: ( 20, 1)( 13, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 68)( 1, 3970)
16863.728 1243.3902 4216 4059  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 70)( 1, 3967)
16864.251 1243.4239 4216 4060  9: ( 21, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 68)( 1, 3970)
D16864.293 1243.4221 4216 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 10)( 2, 73)( 1, 3966)  derive 7
16864.427 1243.4359 4216 4060  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 68)( 1, 3970)
16864.518 1243.4413 4216 4059  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 12)( 2, 72)( 1, 3966)
16864.649 1243.4495 4216 4060  9: ( 20, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 68)( 1, 3970)
D16864.694 1243.4501 4216 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 12)( 2, 72)( 1, 3966)  derive 6
16864.714 1243.4549 4216 4059  9: ( 20, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 70)( 1, 3967)
16864.884 1243.4639 4216 4060  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 14)( 2, 67)( 1, 3970)
16864.89 1243.4659 4216 4059  9: ( 19, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 70)( 1, 3967)
D16865.043 1243.4678 4216 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 1)( 4, 5)( 3, 11)( 2, 72)( 1, 3966)  derive 5
16865.111 1243.4794 4216 4059  9: ( 19, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 70)( 1, 3967)
D16865.312 1243.489 4216 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 6, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966)  derive 4
D16865.458 1243.4958 4216 4059  8: ( 19, 1)( 14, 1)( 7, 2)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966)  derive 3
16865.458 1243.5031 4216 4060  9: ( 21, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 69)( 1, 3969)
16865.634 1243.5151 4216 4060  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 69)( 1, 3969)
D16865.68 1243.517 4216 4059  9: ( 19, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966)  derive 2
D16865.856 1243.5247 4216 4059  9: ( 18, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966)  derive 1
16865.856 1243.5287 4216 4060  9: ( 20, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 69)( 1, 3969)
16866.619 1243.5798 4216 4060  9: ( 20, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 69)( 1, 3969)
16867.017 1243.6044 4216 4060  9: ( 19, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 69)( 1, 3969)
16867.375 1243.628 4216 4061  9: ( 21, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 68)( 1, 3971)
16867.551 1243.64 4216 4061  9: ( 20, 1)( 13, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 68)( 1, 3971)
16867.772 1243.6536 4216 4061  9: ( 20, 1)( 12, 1)( 9, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 68)( 1, 3971)
16867.837 1243.659 4216 4060  9: ( 20, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 70)( 1, 3968)
16868.536 1243.7048 4216 4061  9: ( 20, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 68)( 1, 3971)
16868.998 1243.7347 4216 4060  9: ( 19, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 12)( 2, 70)( 1, 3968)
16869.743 1243.7839 4216 4061  9: ( 20, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 69)( 1, 3970)
16871.66 1243.9089 4216 4062  9: ( 20, 1)( 12, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 3)( 3, 13)( 2, 68)( 1, 3972)


16866.157 1243.547 4217 4059  9: ( 19, 1)( 14, 1)( 8, 1)( 7, 1)( 5, 2)( 4, 4)( 3, 11)( 2, 72)( 1, 3966) original 

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