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I have invented two sets of positive integers: highly regular numbers and superior highly regular numbers.

A positive integer $m \leq n$ is a regular of the positive integer $n$ if all prime numbers that divide $m$ also divide $n$.

So for example:

  • 4 has 3 regulars: 1, 2, 4.
  • 6 has 5 regulars: 1, 2, 3, 4, 6.
  • 8 has 4 regulars: 1, 2, 4, 8.
  • 9 has 3 regulars: 1, 3, 9.
  • 10 has 6 regulars: 1, 2, 4, 5, 8, 10.
  • 12 has 8 regulars: 1, 2, 3, 4, 6, 8, 9, 12.
  • A prime number has 2 regulars: 1 and itself.

$r(n)$ denotes the function which counts the quantity of regulars of the positive integer $n$.

A highly regular number is a positive integer $n$ for which $r(n) > r(q)$ for all positive integer $q < n$.

So a highly regular number is a positive integer with more regulars than any smaller positive integer.

I’ve found that the first highly regular numbers up to 210 are: 2, 4, 6, 10, 12, 18, 24, 30, 42, 60, 84, 90, 120, 150, 180, 210.

A superior highly regular number is a positive integer $n$ for which there is an $\epsilon > 0$ such that: $$\frac{r(n)}{n^\epsilon} \geq \frac{r(k)}{k^\epsilon}$$ for all positive integers $k > 1$.

I've found that the first superior highly regular numbers up to 210 are: 2, 6, 30, 210.

Consequently, I’ve made a conjecture: Every primorial is a superior highly regular number, and every superior highly regular number is a primorial.

Could someone prove that my conjecture is true?

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  • $\begingroup$ What are the values for $\epsilon$ that fit $2,6,30,210?$ $\endgroup$
    – Will Jagy
    May 23, 2014 at 1:14
  • $\begingroup$ Oh, well. One thing you do not appear to have done is prove that your $r(n)/n^\epsilon \rightarrow 0$ as $n \rightarrow \infty.$ This was straightforward for Ramanujan, but your function is not number-theoretic multiplicative. $\endgroup$
    – Will Jagy
    May 23, 2014 at 1:32
  • $\begingroup$ - For 2: 1.000 to 0.834 - For 6: 0.834 to 0.796 - For 30: 0.796 to 0.683 - For 210: 0.683 to 0.595 - For 2310: 0.595 to 0.550 - For 30030: 0.550 to 0.504 - For 510510: 0.504 to 0.481 - For 9699690: 0.481 to 0.455 - For 223092870: 0.455 to 0.427 - For 6469693230: 0.426 to 0.416 - For 200560490130: 0.416 to 0.397 - For 7420738134810: 0.397 to 0.386 - For 304250263527210: 0.386 to 0.379 The values I’ve found may be false if my conjecture is false and there exist superior highly regular numbers which are not primorials. $\endgroup$
    – Omega Force
    May 23, 2014 at 1:44
  • $\begingroup$ The regulars of $n$ are the factors of $n$ plus the cases where a prime power factor can have the power increased by $1$ and still stay less than $n$. So for $120=2^3\cdot3 \cdot 5$ the regulars have all the factors plus $16,9,25$. For a number with lots of prime factors, it will be the number of factors plus the number of primes that divide it. I am not sure it brings much to the table beyond what the divisor count does. $\endgroup$
    – Ross Millikan
    May 25, 2014 at 2:07

1 Answer 1

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There is a strong tendency for numbers that just increase the count of "regulars" to resemble primorials.

The output supports some simple ideas, some of which might be provable. 
The primorial numbers, marked with a P, give both a large count "c"
and a small log ratio log n / log c.  Several numbers before a primorial
 are in arithmetic progression, the common difference being the previous
 primorial.  The next few numbers after a primorial are also squarefree; 
there is a relatively large gap in n immediately following a primorial.
Indeed, the next n after a primorial seems to be the result of replacing
 the final prime by its successor, so the jump in magnitude of n is 
entirely dependent on the size of the "prime gap."  And, this next n is,
accordingly, rather larger than the primorial plus the previous primorial. 


log(n)/log(c)   n        c           n = factored
-----           1        1           1 = 1
1               2 P      2           2 = 2
1.2618          4        3           4 = 2^2
1.1132          6 P      5           6 = 2 * 3
1.2850         10        6          10 = 2 * 5
1.1949         12        8          12 = 2^2 * 3
1.2552         18       10          18 = 2 * 3^2
1.3253         24       11          24 = 2^3 * 3
1.1767         30 P     18          30 = 2 * 3 * 5
1.2693         42       19          42 = 2 * 3 * 7
1.2566         60       26          60 = 2^2 * 3 * 5
1.3296         84       28          84 = 2^2 * 3 * 7
1.2983         90       32          90 = 2 * 3^2 * 5
1.3359        120       36         120 = 2^3 * 3 * 5
1.3492        150       41         150 = 2 * 3 * 5^2
1.3722        180       44         180 = 2^2 * 3^2 * 5
1.2672        210 P     68         210 = 2 * 3 * 5 * 7
1.3350        330       77         330 = 2 * 3 * 5 * 11
1.3615        390       80         390 = 2 * 3 * 5 * 13
1.3233        420       96         420 = 2^2 * 3 * 5 * 7
1.3584        630      115         630 = 2 * 3^2 * 5 * 7
1.3811        840      131         840 = 2^3 * 3 * 5 * 7
1.3978       1050      145        1050 = 2 * 3 * 5^2 * 7
1.4136       1260      156        1260 = 2^2 * 3^2 * 5 * 7
1.4266       1470      166        1470 = 2 * 3 * 5 * 7^2
1.4395       1680      174        1680 = 2^4 * 3 * 5 * 7
1.4481       1890      183        1890 = 2 * 3^3 * 5 * 7
1.4550       2100      192        2100 = 2^2 * 3 * 5^2 * 7
1.3719       2310 P    283        2310 = 2 * 3 * 5 * 7 * 11
1.3912       2730      295        2730 = 2 * 3 * 5 * 7 * 13
1.4236       3570      313        3570 = 2 * 3 * 5 * 7 * 17
1.4358       3990      322        3990 = 2 * 3 * 5 * 7 * 19
1.4192       4620      382        4620 = 2^2 * 3 * 5 * 7 * 11
1.4392       5460      395        5460 = 2^2 * 3 * 5 * 7 * 13
1.4465       6930      452        6930 = 2 * 3^2 * 5 * 7 * 11
1.4680       8190      463        8190 = 2 * 3^2 * 5 * 7 * 13
1.4669       9240      505        9240 = 2^3 * 3 * 5 * 7 * 11
1.4872      10920      519       10920 = 2^3 * 3 * 5 * 7 * 13
1.4820      11550      551       11550 = 2 * 3 * 5^2 * 7 * 11
1.5017      13650      567       13650 = 2 * 3 * 5^2 * 7 * 13
1.4935      13860      593       13860 = 2^2 * 3^2 * 5 * 7 * 11
1.5038      16170      629       16170 = 2 * 3 * 5 * 7^2 * 11
1.5132      18480      660       18480 = 2^4 * 3 * 5 * 7 * 11
1.5206      20790      691       20790 = 2 * 3^3 * 5 * 7 * 11
1.5281      23100      717       23100 = 2^2 * 3 * 5^2 * 7 * 11
1.5343      25410      743       25410 = 2 * 3 * 5 * 7 * 11^2
1.5403      27720      766       27720 = 2^3 * 3^2 * 5 * 7 * 11
1.4609      30030 P   1161       30030 = 2 * 3 * 5 * 7 * 11 * 13
1.4878      39270     1224       39270 = 2 * 3 * 5 * 7 * 11 * 17
1.4985      43890     1253       43890 = 2 * 3 * 5 * 7 * 11 * 19
1.5056      46410     1257       46410 = 2 * 3 * 5 * 7 * 13 * 17
1.5165      51870     1285       51870 = 2 * 3 * 5 * 7 * 13 * 19
1.5165      53130     1306       53130 = 2 * 3 * 5 * 7 * 11 * 23
1.5010      60060     1526       60060 = 2^2 * 3 * 5 * 7 * 11 * 13
1.5281      78540     1597       78540 = 2^2 * 3 * 5 * 7 * 11 * 17
1.5388      87780     1631       87780 = 2^2 * 3 * 5 * 7 * 11 * 19
1.5244      90090     1779       90090 = 2 * 3^2 * 5 * 7 * 11 * 13
1.5514     117810     1856      117810 = 2 * 3^2 * 5 * 7 * 11 * 17
1.5411     120120     1977      120120 = 2^3 * 3 * 5 * 7 * 11 * 13
1.5539     150150     2144      150150 = 2 * 3 * 5^2 * 7 * 11 * 13
1.5639     180180     2294      180180 = 2^2 * 3^2 * 5 * 7 * 11 * 13
1.5729     210210     2420      210210 = 2 * 3 * 5 * 7^2 * 11 * 13
1.5804     240240     2538      240240 = 2^4 * 3 * 5 * 7 * 11 * 13
1.5871     270270     2645      270270 = 2 * 3^3 * 5 * 7 * 11 * 13
1.5931     300300     2743      300300 = 2^2 * 3 * 5^2 * 7 * 11 * 13
1.5984     330330     2836      330330 = 2 * 3 * 5 * 7 * 11^2 * 13
1.6034     360360     2921      360360 = 2^3 * 3^2 * 5 * 7 * 11 * 13
1.6080     390390     3001      390390 = 2 * 3 * 5 * 7 * 11 * 13^2
1.6120     420420     3080      420420 = 2^2 * 3 * 5 * 7^2 * 11 * 13
1.6159     450450     3153      450450 = 2 * 3^2 * 5^2 * 7 * 11 * 13
1.6195     480480     3223      480480 = 2^5 * 3 * 5 * 7 * 11 * 13
1.5489     510510 P   4843      510510 = 2 * 3 * 5 * 7 * 11 * 13 * 17
1.5584     570570     4939      570570 = 2 * 3 * 5 * 7 * 11 * 13 * 19
1.5742     690690     5119      690690 = 2 * 3 * 5 * 7 * 11 * 13 * 23
1.5826     746130     5138      746130 = 2 * 3 * 5 * 7 * 11 * 17 * 19
1.5926     870870     5364      870870 = 2 * 3 * 5 * 7 * 11 * 13 * 29
1.5979     930930     5436      930930 = 2 * 3 * 5 * 7 * 11 * 13 * 31
1.5837    1021020     6225     1021020 = 2^2 * 3 * 5 * 7 * 11 * 13 * 17
1.5932    1141140     6337     1141140 = 2^2 * 3 * 5 * 7 * 11 * 13 * 19
1.6091    1381380     6546     1381380 = 2^2 * 3 * 5 * 7 * 11 * 13 * 23
1.6175    1492260     6560     1492260 = 2^2 * 3 * 5 * 7 * 11 * 17 * 19
1.6040    1531530     7178     1531530 = 2 * 3^2 * 5 * 7 * 11 * 13 * 17
1.6135    1711710     7299     1711710 = 2 * 3^2 * 5 * 7 * 11 * 13 * 19
1.6183    2042040     7928     2042040 = 2^3 * 3 * 5 * 7 * 11 * 13 * 17
1.6278    2282280     8055     2282280 = 2^3 * 3 * 5 * 7 * 11 * 13 * 19
1.6293    2552550     8553     2552550 = 2 * 3 * 5^2 * 7 * 11 * 13 * 17
1.6389    2852850     8685     2852850 = 2 * 3 * 5^2 * 7 * 11 * 13 * 19
1.6383    3063060     9099     3063060 = 2^2 * 3^2 * 5 * 7 * 11 * 13 * 17
1.6478    3423420     9236     3423420 = 2^2 * 3^2 * 5 * 7 * 11 * 13 * 19
1.6459    3573570     9580     3573570 = 2 * 3 * 5 * 7^2 * 11 * 13 * 17
1.6554    3993990     9719     3993990 = 2 * 3 * 5 * 7^2 * 11 * 13 * 19
1.6525    4084080    10010     4084080 = 2^4 * 3 * 5 * 7 * 11 * 13 * 17
1.6620    4564560    10155     4564560 = 2^4 * 3 * 5 * 7 * 11 * 13 * 19
1.6582    4594590    10414     4594590 = 2 * 3^3 * 5 * 7 * 11 * 13 * 17
1.6634    5105100    10777     5105100 = 2^2 * 3 * 5^2 * 7 * 11 * 13 * 17
1.6681    5615610    11120     5615610 = 2 * 3 * 5 * 7 * 11^2 * 13 * 17
1.6723    6126120    11441     6126120 = 2^3 * 3^2 * 5 * 7 * 11 * 13 * 17
1.6762    6636630    11740     6636630 = 2 * 3 * 5 * 7 * 11 * 13^2 * 17
1.6798    7147140    12027     7147140 = 2^2 * 3 * 5 * 7^2 * 11 * 13 * 17
1.6832    7657650    12293     7657650 = 2 * 3^2 * 5^2 * 7 * 11 * 13 * 17
1.6864    8168160    12549     8168160 = 2^5 * 3 * 5 * 7 * 11 * 13 * 17
1.6893    8678670    12799     8678670 = 2 * 3 * 5 * 7 * 11 * 13 * 17^2
1.6920    9189180    13037     9189180 = 2^2 * 3^3 * 5 * 7 * 11 * 13 * 17
1.6245    9699690 P  19985     9699690 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19  
1.6387   11741730    20605    11741730 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 23
1.6474   13123110    20929    13123110 = 2 * 3 * 5 * 7 * 11 * 13 * 19 * 23
1.6554   14804790    21453    14804790 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 29
1.6600   15825810    21713    15825810 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 31
1.6641   16546530    21769    16546530 = 2 * 3 * 5 * 7 * 11 * 13 * 19 * 29
1.6688   17687670    22028    17687670 = 2 * 3 * 5 * 7 * 11 * 13 * 19 * 31
1.6722   18888870    22443    18888870 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 37
1.6552   19399380    25289    19399380 = 2^2 * 3 * 5 * 7 * 11 * 13 * 17 * 19
1.6694   23483460    26005    23483460 = 2^2 * 3 * 5 * 7 * 11 * 13 * 17 * 23
1.6781   26246220    26370    26246220 = 2^2 * 3 * 5 * 7 * 11 * 13 * 19 * 23
1.6730   29099070    28924    29099070 = 2 * 3^2 * 5 * 7 * 11 * 13 * 17 * 19
1.6872   35225190    29701    35225190 = 2 * 3^2 * 5 * 7 * 11 * 13 * 17 * 23
1.6856   38798760    31776    38798760 = 2^3 * 3 * 5 * 7 * 11 * 13 * 17 * 19
1.6998   46966920    32594    46966920 = 2^3 * 3 * 5 * 7 * 11 * 13 * 17 * 23
1.6953   48498450    34150    48498450 = 2 * 3 * 5^2 * 7 * 11 * 13 * 17 * 19
1.7032   58198140    36204    58198140 = 2^2 * 3^2 * 5 * 7 * 11 * 13 * 17 * 19
1.7099   67897830    38028    67897830 = 2 * 3 * 5 * 7^2 * 11 * 13 * 17 * 19
1.7157   77597520    39660    77597520 = 2^4 * 3 * 5 * 7 * 11 * 13 * 17 * 19
1.7208   87297210    41161    87297210 = 2 * 3^3 * 5 * 7 * 11 * 13 * 17 * 19
1.7254   96996900    42543    96996900 = 2^2 * 3 * 5^2 * 7 * 11 * 13 * 17 * 19
1.7295  106696590    43827   106696590 = 2 * 3 * 5 * 7 * 11^2 * 13 * 17 * 19
1.7333  116396280    45029   116396280 = 2^3 * 3^2 * 5 * 7 * 11 * 13 * 17 * 19
1.7367  126095970    46156   126095970 = 2 * 3 * 5 * 7 * 11 * 13^2 * 17 * 19
1.7399  135795660    47233   135795660 = 2^2 * 3 * 5 * 7^2 * 11 * 13 * 17 * 19
1.7429  145495350    48240   145495350 = 2 * 3^2 * 5^2 * 7 * 11 * 13 * 17 * 19
1.7457  155195040    49202   155195040 = 2^5 * 3 * 5 * 7 * 11 * 13 * 17 * 19
1.7483  164894730    50130   164894730 = 2 * 3 * 5 * 7 * 11 * 13 * 17^2 * 19
1.7507  174594420    51014   174594420 = 2^2 * 3^3 * 5 * 7 * 11 * 13 * 17 * 19
1.7530  184294110    51861   184294110 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19^2
1.7552  193993800    52680   193993800 = 2^3 * 3 * 5^2 * 7 * 11 * 13 * 17 * 19
1.7573  203693490    53468   203693490 = 2 * 3^2 * 5 * 7^2 * 11 * 13 * 17 * 19
1.7593  213393180    54226   213393180 = 2^2 * 3 * 5 * 7 * 11^2 * 13 * 17 * 19
1.6970  223092870 P  83074   223092870 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23
1.7121  281291010    86054   281291010 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 29
1.7164  300690390    86978   300690390 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 31
log(n)/log(c)  n       c           n   =   factored

set<int> PrimeFac(int i)
{
  set<int> boo;

  int p = 2;
  int temp = i;
  if (temp < 0 )
  {
    temp *= -1;
  }

  if ( temp > 1)
  {

    while( temp > 1 && p * p <= temp)
    {
      if (temp % p == 0)
      {

        boo.insert(p);

        temp /= p;

        while (temp % p == 0)
        {
          temp /= p;

        } // while p is fac

      }  // if p is factor
      ++p;
    } // while p

    if (temp > 1 )  boo.insert(temp);
  } // temp > 1
  return boo;
} // end of  PrimeFac



set<int> Radical_Divides_Radical(int n)
{
   set<int> boo;
   boo.insert(1);
   if (n <= 1) return boo;
   else
   {
     set<int> fax;
     fax = PrimeFac(n);
     set<int>::iterator iter;

     for(iter = fax.begin() ;  iter != fax.end(); ++iter)
     {
        set<int> more;
       int p = *iter;
       int power = 1;
    set<int>::iterator iter2;
       while ( power <= n / p)
       {
         power *= p;

          for(iter2 = boo.begin() ;  iter2 != boo.end(); ++iter2)
          {
            int u = *iter2;
            if ( u <= n / power ) more.insert( power * u);
          } // iter 2
       } // while
        for(iter2 = more.begin() ;  iter2 != more.end(); ++iter2)
          {
            int u = *iter2;
            if ( u <= n ) boo.insert(  u);
          } // iter 2
     } // iter
     return boo;
   } // else   n >= 2

} // end of Radical_Divides_Radical
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  • $\begingroup$ What language did you use for this? I know some number libraries that could really speed this up (numbers here are quite small) $\endgroup$
    – qwr
    May 24, 2014 at 4:39
  • $\begingroup$ @qwr, it's just C++, in this case ordinary integers, so it cannot go more than $2^{31}.$ The summary is that i do not see how the analogy with Ramanujan's method is going to work, too many aspects are not proved. $\endgroup$
    – Will Jagy
    May 24, 2014 at 18:12
  • $\begingroup$ @qwr, edited in the two main routines used. this is C++ with STL. $\endgroup$
    – Will Jagy
    May 24, 2014 at 20:17

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