Find a counterexample: For every antiprime $n>1$, there is a prime divisor $p$ such that $n/p$ is an antiprime

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?.


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)

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.

Tags:

Number Theory

Elementary Number Theory

Divisibility

Examples Counterexamples

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