User:Proofname123/Sandbox/Primenumber growth within exponential gaps

Some reseach I did on wikipedia and the next findings can resemble some other conjectures or mathematical functions.

For instance Chebyshev function. source=> https://en.wikipedia.org/wiki/Chebyshev_function#The_exact_formula Or the Goldbach conjecture. source=> https://en.wikipedia.org/wiki/Goldbach%27s_conjecture

1. What is the formula about?

2. the formula has twoo components.

3. The formula.

4. The first hundred results.

5. Does this formula already excists?

6. The programm in c++

7. resemblence with other formulas and the meaning of this formula.

1. What is the formula about?

In this formula, what might be seen as a conjecture, some findings resemble an evenly growth of primenumbers within an exponential growth of natural numbers.

2. the formula has twoo components.

The formula has twoo components.

- the growth of natural number gaps expanding exponential - the growth of a quantity of primenumbers within the gaps as ment above

3. The formula.

1²/1 = 1     3²/2 = 4.5     6²/3 = 12    10²/4 = 25    ...

the input is 1² as an input of a triangular number so in the second the input is 3. Divided by a consecutive number adding from 1 + 1 etc.  So  triangular number ²/ natural consecutive number

In between those results, a gap, there is an amount of primenumbers. A substantial growth is seen in the results of the execution of a c++ programm.

4. the first 99 resulst. These resulst come from a c++ program.

number of gap | amount of primenumbers | the gaps

1:  1   0 <=> 1                             1         per gap=>             100 %           total=>      50 % 2:  2   1 <=> 4.5                           1         per gap=>             57.1429 %       total=>      85.7143 % 3:  3   4.5 <=> 12                          1         per gap=>             40 %            total=>      52.8571 % 4:  4   12 <=> 25                           1         per gap=>             30.7692 %       total=>      38.1538 % 5:  5   25 <=> 45                           1         per gap=>             25 %            total=>      29.8077 % 6:  7   45 <=> 73.5                         1.16667   per gap=>             24.5614 %       total=>      24.9373 % 7:  9   73.5 <=> 112                        1.28571   per gap=>             23.3766 %       total=>      24.4133 % 8:  8   112 <=> 162                         1         per gap=>             16 %            total=>      22.557 % 9:  11   162 <=> 225                        1.22222   per gap=>             17.4603 %       total=>      16.146 % 10:  14   225 <=> 302.5                     1.4       per gap=>             18.0645 %       total=>      17.5152 % 11:  15   302.5 <=> 396           1.36364         per gap=>             16.0428 %       total=>      17.896 % 12:  19   396 <=> 507           1.58333         per gap=>             17.1171 %       total=>      16.1254 % 13:  19   507 <=> 637           1.46154         per gap=>             14.6154 %       total=>      16.9384 % 14:  23   637 <=> 787.5           1.64286         per gap=>             15.2824 %       total=>      14.6599 % 15:  25   787.5 <=> 960           1.66667         per gap=>             14.4928 %       total=>      15.233 % 16:  29   960 <=> 1156           1.8125         per gap=>             14.7959 %       total=>      14.5106 % 17:  29   1156 <=> 1377           1.70588         per gap=>             13.1222 %       total=>      14.7029 % 18:  37   1377 <=> 1624.5           2.05556         per gap=>             14.9495 %       total=>      13.2183 % 19:  33   1624.5 <=> 1900           1.73684         per gap=>             11.9782 %       total=>      14.8009 % 20:  38   1900 <=> 2205           1.9         per gap=>             12.459 %       total=>      12.0011 % 21:  43   2205 <=> 2541           2.04762         per gap=>             12.7976 %       total=>      12.4744 % 22:  50   2541 <=> 2909.5           2.27273         per gap=>             13.5685 %       total=>      12.8311 % 23:  45   2909.5 <=> 3312           1.95652         per gap=>             11.1801 %       total=>      13.469 % 24:  57   3312 <=> 3750           2.375         per gap=>             13.0137 %       total=>      11.2535 % 25:  56   3750 <=> 4225           2.24         per gap=>             11.7895 %       total=>      12.9666 % 26:  61   4225 <=> 4738.5           2.34615         per gap=>             11.8793 %       total=>      11.7928 % 27:  62   4738.5 <=> 5292           2.2963         per gap=>             11.2014 %       total=>      11.8551 % 28:  74   5292 <=> 5887           2.64286         per gap=>             12.437 %       total=>      11.244 % 29:  68   5887 <=> 6525           2.34483         per gap=>             10.6583 %       total=>      12.3777 % 30:  77   6525 <=> 7207.5           2.56667         per gap=>             11.2821 %       total=>      10.6784 % 31:  83   7207.5 <=> 7936           2.67742         per gap=>             11.3933 %       total=>      11.2855 % 32:  83   7936 <=> 8712           2.59375         per gap=>             10.6959 %       total=>      11.3721 % 33:  95   8712 <=> 9537           2.87879         per gap=>             11.5152 %       total=>      10.72 % 34:  94   9537 <=> 10412.5           2.76471         per gap=>             10.7367 %       total=>      11.4929 % 35:  96   10412.5 <=> 11340           2.74286         per gap=>             10.3504 %       total=>      10.726 % 36:  101   11340 <=> 12321           2.80556         per gap=>             10.2956 %       total=>      10.3489 % 37:  114   12321 <=> 13357           3.08108         per gap=>             11.0039 %       total=>      10.3143 % 38:  110   13357 <=> 14449.5           2.89474         per gap=>             10.0686 %       total=>      10.9799 % 39:  124   14449.5 <=> 15600           3.17949         per gap=>             10.7779 %       total=>      10.0864 % 40:  121   15600 <=> 16810           3.025         per gap=>             10 %       total=>      10.7589 % 41:  133   16810 <=> 18081           3.2439         per gap=>             10.4642 %       total=>      10.0111 % 42:  125   18081 <=> 19414.5           2.97619         per gap=>             9.37383 %       total=>      10.4388 % 43:  147   19414.5 <=> 20812           3.4186         per gap=>             10.5188 %       total=>      9.39985 % 44:  150   20812 <=> 22275           3.40909         per gap=>             10.2529 %       total=>      10.5129 % 45:  153   22275 <=> 23805           3.4         per gap=>             10 %       total=>      10.2474 % 46:  153   23805 <=> 25403.5           3.32609         per gap=>             9.57147 %       total=>      9.99088 % 47:  168   25403.5 <=> 27072           3.57447         per gap=>             10.0689 %       total=>      9.58184 % 48:  169   27072 <=> 28812           3.52083         per gap=>             9.71264 %       total=>      10.0617 % 49:  165   28812 <=> 30625           3.36735         per gap=>             9.10094 %       total=>      9.70041 % 50:  187   30625 <=> 32512.5           3.74         per gap=>             9.90728 %       total=>      9.11675 % 51:  193   32512.5 <=> 34476           3.78431         per gap=>             9.82939 %       total=>      9.90579 % 52:  188   34476 <=> 36517           3.61538         per gap=>             9.21117 %       total=>      9.81772 % 53:  199   36517 <=> 38637           3.75472         per gap=>             9.38679 %       total=>      9.21442 % 54:  206   38637 <=> 40837.5           3.81481         per gap=>             9.36151 %       total=>      9.38633 % 55:  230   40837.5 <=> 43120           4.18182         per gap=>             10.0767 %       total=>      9.37428 % 56:  210   43120 <=> 45486           3.75         per gap=>             8.87574 %       total=>      10.0556 % 57:  224   45486 <=> 47937           3.92982         per gap=>             9.13913 %       total=>      8.88028 % 58:  239   47937 <=> 50474.5           4.12069         per gap=>             9.41872 %       total=>      9.14387 % 59:  239   50474.5 <=> 53100           4.05085         per gap=>             9.10303 %       total=>      9.41346 % 60:  246   53100 <=> 55815           4.1         per gap=>             9.06077 %       total=>      9.10234 % 61:  269   55815 <=> 58621           4.40984         per gap=>             9.5866 %       total=>      9.06925 % 62:  257   58621 <=> 61519.5           4.14516         per gap=>             8.86666 %       total=>      9.57517 % 63:  265   61519.5 <=> 64512           4.20635         per gap=>             8.85547 %       total=>      8.86648 % 64:  282   64512 <=> 67600           4.40625         per gap=>             9.13212 %       total=>      8.85973 % 65:  274   67600 <=> 70785           4.21538         per gap=>             8.60283 %       total=>      9.1241 % 66:  297   70785 <=> 74068.5           4.5         per gap=>             9.04523 %       total=>      8.60943 % 67:  302   74068.5 <=> 77452           4.50746         per gap=>             8.92567 %       total=>      9.04347 % 68:  314   77452 <=> 80937           4.61765         per gap=>             9.01004 %       total=>      8.92689 % 69:  319   80937 <=> 84525           4.62319         per gap=>             8.89075 %       total=>      9.00834 % 70:  315   84525 <=> 88217.5           4.5         per gap=>             8.53081 %       total=>      8.88568 % 71:  333   88217.5 <=> 92016           4.69014         per gap=>             8.76662 %       total=>      8.53408 % 72:  355   92016 <=> 95922           4.93056         per gap=>             9.08858 %       total=>      8.77103 % 73:  344   95922 <=> 99937           4.71233         per gap=>             8.56787 %       total=>      9.08155 % 74:  352   99937 <=> 104062           4.75676         per gap=>             8.5323 %       total=>      8.5674 % 75:  364   104062 <=> 108300           4.85333         per gap=>             8.58997 %       total=>      8.53306 % 76:  371   108300 <=> 112651           4.88158         per gap=>             8.52678 %       total=>      8.58915 % 77:  379   112651 <=> 117117           4.92208         per gap=>             8.48634 %       total=>      8.52626 % 78:  400   117117 <=> 121700           5.12821         per gap=>             8.72886 %       total=>      8.48941 % 79:  400   121700 <=> 126400           5.06329         per gap=>             8.50973 %       total=>      8.72612 % 80:  406   126400 <=> 131220           5.075         per gap=>             8.42324 %       total=>      8.50867 % 81:  417   131220 <=> 136161           5.14815         per gap=>             8.43959 %       total=>      8.42344 % 82:  438   136161 <=> 141224           5.34146         per gap=>             8.65014 %       total=>      8.44212 % 83:  429   141224 <=> 146412           5.16867         per gap=>             8.26988 %       total=>      8.64562 % 84:  457   146412 <=> 151725           5.44048         per gap=>             8.60154 %       total=>      8.27378 % 85:  447   151725 <=> 157165           5.25882         per gap=>             8.21691 %       total=>      8.59707 % 86:  461   157165 <=> 162734           5.36047         per gap=>             8.27871 %       total=>      8.21762 % 87:  458   162734 <=> 168432           5.26437         per gap=>             8.0372 %       total=>      8.27597 % 88:  489   168432 <=> 174262           5.55682         per gap=>             8.38765 %       total=>      8.04114 % 89:  501   174262 <=> 180225           5.62921         per gap=>             8.40181 %       total=>      8.38781 % 90:  511   180225 <=> 186322           5.67778         per gap=>             8.38048 %       total=>      8.40158 % 91:  505   186322 <=> 192556           5.54945         per gap=>             8.10139 %       total=>      8.37745 % 92:  524   192556 <=> 198927           5.69565         per gap=>             8.22477 %       total=>      8.10271 % 93:  522   198927 <=> 205437           5.6129         per gap=>             8.01843 %       total=>      8.22257 % 94:  562   205437 <=> 212088           5.97872         per gap=>             8.45049 %       total=>      8.02298 % 95:  536   212088 <=> 218880           5.64211         per gap=>             7.89106 %       total=>      8.44466 % 96:  572   218880 <=> 225816           5.95833         per gap=>             8.24683 %       total=>      7.89472 % 97:  579   225816 <=> 232897           5.96907         per gap=>             8.17681 %       total=>      8.24611 % 98:  566   232897 <=> 240124           5.77551         per gap=>             7.8312 %       total=>      8.17332 % 99:  597   240124 <=> 247500           6.0303         per gap=>             8.09437 %       total=>      7.83383 %

You might aswell read the other resulst in percentages wich tells about the growth. I think these results are somewhat spectaculair to see a growth of primenumbers wich sort of evenly grow.

5. Does this formula already excists?

I don't know enough about this and I would like to know. This might resemble other functions or conjectures.

6. The programm in c++ (program language)


 * 1) include
 * 2) include
 * 3) include

using namespace std;

int main {   int count, n, c;	double x = 0; double b = 0; double bnext = 0; double z = 1; double y = 0; double l = 0; double h = 0; double m = 0; double k = 0; double v = 0; while(x < 10000) { y = y + z;		z = z + 1.5; bnext = b;	   b = b + y;	    l = b *1; h = bnext *1; x++; for (n = h; n <= l; n++){ count = 0; for (int i = 2; i <= n/2; i++){ if(n%i==0){ count++; break; }       }        if(count==0 && n!=1){ c = c + 1; }   }	k = m;	m = (c/(l-h))*100; v = ((k*x) +m)/(x+1); cout << x   << ":   "  << c  << "   " << h << " <=> " << l << "           " << c/x  << "         per gap=>             " <<  m << " %" << "       total=>      "  << v  << " %"  << endl;

c= 0; }	return 0; }

7. resemblance with other formulas and the meaning of this formula.

The question is what this formula says to me and if there are already such formulas that say about thesame thing. To me this formula says that there is an analyse for to track the growth of primenumbers. I'm curious for more results. I came up with it, with considering there are no more numbers than an amount of numbers. Like saying 1, 2, 3, 4 that's all. for instance gap one: here only number one is used         (1 x 1 = 1) is also 1²                                                                          divided by one here only number one and twoo is used (1 x 1 = 1 plus 1 x 2 = 2 plus 2 x 1 = 2  plus 2 x 2 = 4) so 1 + 2 + 2 + 4 = 9  is also 3²     divided by twoo etc.