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]]>7# = 7 x 5 x 3 x 2

There are a multitude of patterns to be found in the primes……..explaining and understanding them all in a comprehensive simple-ish system is the challenge

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]]>I charted these numbers on the Sacks Spiral and drew lines between them – with my limited tools, it looks like these make angles of 30 degrees each turn, though perhaps a little wider for for 5-10-15.

Do you know if the numbers where symmetry fails to be prime (57, 77, 129, 141, 143, 169…) or their would-be counterpart primes (43, 73, 139, 137 151,191…) have any similar properties?

Also, I’ve noticed that the primes near the 5# seem to fall in a bit of a sin wave pattern, though this could just be an illusion that falls apart once I’ve plotted further out.

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]]>Do you know how to make the Fibonacci numbers? 1+1 = 2, 1+2 = 3, 2+3 = 5, 3+5 = 8 and so on. Then do the Tribonacci numbers 1+1+1 = 3, 1+1+3 = 5, 1+3+5 = 9 and so on. Continue like that to infinity, Tetranacci, Pentanacci, n-anacci.

Now instead of adding to positive number, wrap the result with minus. -(1+1) = -2, -(1+-2)= 1, -(-2+1) = 1, -(1+1) = -2, and so on. Do the same for Triboacci -(1+1+1) = -3, -(1+1+-3) = 1, and so on.

Now make it more complicated, put all these NEGATIVE n-anacci numbers next to each other with one column for each n-anacci number. That is, the negative Fibonacci numbers in the third column, the negative tribonacci numbers in the fourth column and so on.

If you divide the result with the row index and sum it up, you will get the natural logarithm of n. That is, for negative Fibonacci numbers you will get the logarithm of 3, for negative Tribonacci numbers you will get logarithm of 4, and so on.

Now make it really complicated. Let the recursions for negative n-anacci numbers that I described, run in cross directions to each other as in here:

http://en.wikipedia.org/wiki/Von_Mangoldt_function#Expansion_of_terms

So that you get matrix T. Then repeat and divide by either the row index or the column index (both row index and column index goes because of the symmetry), sum it up and you have the von Mangoldt function which is the fundamental function in the fundamental theorem of arithmetic, next after the Möbius function and the Dirichlet inverse of the Euler totient function.

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]]>See this link: https://oeis.org/history?seq=A225776 and click through the “Older changes” for the exposition of a sieve based on digital roots. Very similar to some of the stuff you and Croft get into on this blog.

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