15 =3*5
21 =3*7
35 =5*7
3 total distinct semi-primes <=35 each prime of 1 length.
--------------------------------------------------------
143 =11*13
... 20 total
187 =11*17
... 19 total
etc.
210 total distinct semi-primes <=8633 each prime of 2 length.
------------------------------------------------------------
10403 =101*103
... 142 total
10807 =101*107
... 141 total
etc.
10153 total distinct semi-primes <=988027 each prime of 3 length.
----------------------------------------------------------------
1022117 =1009*1013
'' 1060 total
1028171 =1009*1019
... 1059 total
ect.
562330 total distinct semi-primes <= 99400891 each of the 2 primes has 4 length.
--------------------------------------------------------------------------------
100160063 =10007*10009
...8362 total
100440259 =10007*10037
...8361 total
etc.
34965703 total distinct semi-primes <= 9998000099 each of the 2 primes has 5 digits.
------------------------------------------------------------------------------------
10002200057=100003*100019
...68855 total
10004600129=100003*100043
...68854 total
etc.
2370539940 total distinct semi-primes <= 1999962000357 each of the 2 primes has 6 digits.
------------------------------------------------------------------------------------------
Using triangle numbers to get the correct distinct number of semi-primes in each group.
I believe this is correct but I could be wrong.

Dan

djoyce099 <hlauk.h.bogart@gmail.com> wrote:
> 15 =3*5
> 21 =3*7
> 35 =5*7
> 3 total distinct semi-primes <=35 each prime of 1 length.

This should be 6. You left out 2*3, 2*5, 2*7.

> --------------------------------------------------------
....
> 210 total distinct semi-primes <=8633 each prime of 2 length.
> ------------------------------------------------------------
....
> 10153 total distinct semi-primes <=988027 each prime of 3 length.
> ----------------------------------------------------------------
....
> 562330 total distinct semi-primes <= 99400891 each of the 2 primes has 4 length.
> --------------------------------------------------------------------------
....
> 34965703 total distinct semi-primes <= 9998000099 each of the 2 primes has 5 digits.
> -----------------------------------------------------------------------
> 10002200057=100003*100019
> ..68855 total
> 10004600129=100003*100043
> ..68854 total
> etc.
> 2370539940 total distinct semi-primes <= 1999962000357 each of the 2 primes has 6 digits.

Wrong number there, if my calculation was correct. I think there are
68906 primes between 100000 and 1000000, rather than 68856. Here's
the bash shell script I used for checking your numbers, followed by
its 7 lines of output, which took 1 second to compute:

d=10; while [ $d -lt 100000000 ]; do c=$(primes $d $((d*10))|wc -l); echo $d $c $(dc -e"$c d1-*2/p"); d=$((d*10)); done
10 21 210
100 143 10153
1000 1061 562330
10000 8363 34965703
100000 68906 2373983965
1000000 586081 171745176240
10000000 5096876 12989069931250

> -------------------------------------------------------------------------
> Using triangle numbers to get the correct distinct number of semi-primes in each group.
> I believe this is correct but I could be wrong.

On Thursday, July 15, 2021 at 2:26:00 AM UTC-4, James Waldby wrote:
> djoyce099 <hlauk.h...@gmail.com> wrote:
> > 15 =3*5
> > 21 =3*7
> > 35 =5*7
> > 3 total distinct semi-primes <=35 each prime of 1 length.
> This should be 6. You left out 2*3, 2*5, 2*7.
>
> > --------------------------------------------------------
> ...
> > 210 total distinct semi-primes <=8633 each prime of 2 length.
> > ------------------------------------------------------------
> ...
> > 10153 total distinct semi-primes <=988027 each prime of 3 length.
> > ----------------------------------------------------------------
> ...
> > 562330 total distinct semi-primes <= 99400891 each of the 2 primes has 4 length.
> > --------------------------------------------------------------------------
> ...
> > 34965703 total distinct semi-primes <= 9998000099 each of the 2 primes has 5 digits.
> > -----------------------------------------------------------------------
> > 10002200057=100003*100019
> > ..68855 total
> > 10004600129=100003*100043
> > ..68854 total
> > etc.
> > 2370539940 total distinct semi-primes <= 1999962000357 each of the 2 primes has 6 digits.
> Wrong number there, if my calculation was correct. I think there are
> 68906 primes between 100000 and 1000000, rather than 68856. Here's
> the bash shell script I used for checking your numbers, followed by
> its 7 lines of output, which took 1 second to compute:
>
> d=10; while [ $d -lt 100000000 ]; do c=$(primes $d $((d*10))|wc -l); echo $d $c $(dc -e"$c d1-*2/p"); d=$((d*10)); done
> 10 21 210
> 100 143 10153
> 1000 1061 562330
> 10000 8363 34965703
> 100000 68906 2373983965
> 1000000 586081 171745176240
> 10000000 5096876 12989069931250
> > -------------------------------------------------------------------------
> > Using triangle numbers to get the correct distinct number of semi-primes in each group.
> > I believe this is correct but I could be wrong.

My OP stated just odd primes.
I get 20 instead of 21 because index t(20) = 210
I get 142 instead of 143 because index t(142) = 10153
I get 1060 instead of 1061 because index t(1060) = 562330
I get 8362 instead of 8363 because index t(8362) = 34965703
Even though the results are the same there are actually only 20 combinations of semi-primes starting with 11*13,
11*17, 11*19 .. ,11*97. So that is why I show (1) less in the next 3.
The last one you are probably right with (6) digits each prime where I show less total semi-primes for that group than you.
I'll have to look at my program.

Thanks for the the extra (7) and (8) digit primes. That last one, (8) digit has almost 13 trillion semi-primes with prime lengths of (8) digit each. A number not even half our national dept, let alone unfunded liabilities which is approaching
100 trillion. Absolutly mind boggolling.

Dan

On Thursday, July 15, 2021 at 3:04:12 PM UTC-4, djoyce099 wrote:
> On Thursday, July 15, 2021 at 2:26:00 AM UTC-4, James Waldby wrote:
> > djoyce099 <hlauk.h...@gmail.com> wrote:
> > > 15 =3*5
> > > 21 =3*7
> > > 35 =5*7
> > > 3 total distinct semi-primes <=35 each prime of 1 length.
> > This should be 6. You left out 2*3, 2*5, 2*7.
> >
> > > --------------------------------------------------------
> > ...
> > > 210 total distinct semi-primes <=8633 each prime of 2 length.
> > > ------------------------------------------------------------
> > ...
> > > 10153 total distinct semi-primes <=988027 each prime of 3 length.
> > > ----------------------------------------------------------------
> > ...
> > > 562330 total distinct semi-primes <= 99400891 each of the 2 primes has 4 length.
> > > --------------------------------------------------------------------------
> > ...
> > > 34965703 total distinct semi-primes <= 9998000099 each of the 2 primes has 5 digits.
> > > -----------------------------------------------------------------------
> > > 10002200057=100003*100019
> > > ..68855 total
> > > 10004600129=100003*100043
> > > ..68854 total
> > > etc.
> > > 2370539940 total distinct semi-primes <= 1999962000357 each of the 2 primes has 6 digits.
> > Wrong number there, if my calculation was correct. I think there are
> > 68906 primes between 100000 and 1000000, rather than 68856. Here's
> > the bash shell script I used for checking your numbers, followed by
> > its 7 lines of output, which took 1 second to compute:
> >
> > d=10; while [ $d -lt 100000000 ]; do c=$(primes $d $((d*10))|wc -l); echo $d $c $(dc -e"$c d1-*2/p"); d=$((d*10)); done
> > 10 21 210
> > 100 143 10153
> > 1000 1061 562330
> > 10000 8363 34965703
> > 100000 68906 2373983965
> > 1000000 586081 171745176240
> > 10000000 5096876 12989069931250
> > > -------------------------------------------------------------------------
> > > Using triangle numbers to get the correct distinct number of semi-primes in each group.
> > > I believe this is correct but I could be wrong.
> My OP stated just odd primes.
> I get 20 instead of 21 because index t(20) = 210
> I get 142 instead of 143 because index t(142) = 10153
> I get 1060 instead of 1061 because index t(1060) = 562330
> I get 8362 instead of 8363 because index t(8362) = 34965703
> Even though the results are the same there are actually only 20 combinations of semi-primes starting with 11*13,
> 11*17, 11*19 .. ,11*97. So that is why I show (1) less in the next 3.
> The last one you are probably right with (6) digits each prime where I show less total semi-primes for that group than you.
> I'll have to look at my program.
>
> Thanks for the the extra (7) and (8) digit primes. That last one, (8) digit has almost 13 trillion semi-primes with prime lengths of (8) digit each. A number not even half our national dept, let alone unfunded liabilities which is approaching
> 100 trillion. Absolutly mind boggolling.
>
> Dan

I must have left that out in my final edit before posting (only odd primes) , my bad because I intended it to be there.

On Wed, 14 Jul 2021 20:48:34 -0700 (PDT), djoyce099
<hlauk.h.bogart@gmail.com> wrote:

>
>15 =3*5
>21 =3*7
>35 =5*7
>3 total distinct semi-primes <=35 each prime of 1 length.

I don't know why you excluded 9 = 3*3 since it is distinct from the
other semi-primes consisting of one digit factors. It changes the
number of semi-primes but not the underlying nature of the count.

>--------------------------------------------------------
>143 =11*13
>.. 20 total
>187 =11*17
>.. 19 total
>etc.
>210 total distinct semi-primes <=8633 each prime of 2 length.
>------------------------------------------------------------

<snip>

>------------------------------------------------------------------------------------------
>Using triangle numbers to get the correct distinct number of semi-primes in each group.
>I believe this is correct but I could be wrong.

For a specific number of digits, there are n primes with that many
digits. Call them p_1, p_2, ..., p_n.

The distinct semi-primes that contain p_1 as a factor all have the
form p_1 * p_i for i going from 2 to n. There are obviously n-1 such
semi-primes.

The distinct semi-primes that contain p_2 as a factor all have the
form p_2 * p_j for j going from 3 to n. There are obviously n-2 such
semi-primes.

Continuing with p_3 etc shows that the total number of semi-primes is
(n-1) + (n-2) + ... + 3 + 2+ 1.
This sum is known to be n * (n-1) / 2. Since this formula is known to
produce the triangle numbers, your belief is verified.

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