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I seemed to started a snowball! Bobbym, your drawing is excellent to illustrate the 'nested hexagons'. Meanwhile, I have analysed 1000 'nested hexagons', which contains 3,003,001 elemental hexagons so that there are 254 prime numbers and the 'hit ratio' is 1/4. Clearly, there is a trend for the denominator to grow from my original estimate of !/3 (for 200 hexagons) up to 1/4 for 1000 hexagons. 4 is a long way from infinity and, not being a mathematician, I would not know how to prove that ultimate convergence will be zero.
EXCEL is wonderful! I have used my equation H=3n2+3n+1 to calculate all the hexagons in the NESTED HEXAGONS and then this algorithm to identify the PRIME NUMBERS found in the H columns:
=IF(B9=2,"Prime",IF(AND(MOD(B9,ROW(INDIRECT("2:"&ROUNDUP(SQRT(B9),0))))<>0),"Prime","Not Prime"))
Place e.g. the algorithm in a cell opposite B9 and write in B9 and then CRL + SHIFT + ENTER in the address box to activate it.
Scroll down to include all the H cells and the list will read off the PRIME cells.
See: http://www.excelexchange.com/prime_number_test.html
I have competed 1000 NESTED HEXAGONS in pretty quick time with 3,003,001 elemental hexagons. The algorithm works up to 268,435,455 – I am NOT going to the limit!!
The PRIME NUMBER (P) hit ratio is steady diminishing and has reached ~1/4, having generated 254 P’s for 1000 NESTED HEXAGONS.
This exercise was fun but of what use?
Hi Relentless
I have not attempted your statistics but we agree in the number of primes up to 200 hexagons. I too have been busy expanding the number of hexagons thinking that the hit ratio might converge on a couple of well known constants! I completed 400 hexagons (range of 481,201) and the prime 'hit ratio' became 1/3.228 (provided I have not missed any of the primes). So, for hexagon numbers (n) I have have these primes (P):
n P
050 026
100 042
150 053
200 071
250 084
300 096
350 111
400 124
Additionally, I assembled this list of primes from https://www.mathsisfun.com/numbers/prime-number-lists.html
Totals Primes Range
9592 9592 000,002-100,000
17984 8392 100,000-200,000
25997 8013 200,000-300,000
33860 7863 300,000-400,000
41538 7678 400,000-500,000
49098 7560 500,000-600,000
56544 7446 600,000-700,000
63951 7407 700,000-800,000
71275 7324 800,000-900,000
78498 7223 900,000-1,000,000
Hi Relentless
Thank you for converting the squared term to superscript but there is no minus sign in the equation
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