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 Special primes (Posted on 2012-02-12)
There are two primes, both below 100, that evenly divide the sum of the primes less than themselves.

Find those two primes.

The only other such prime below 2,000,000 is M, the n-th prime.

Find M and n.

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 computer solution | Comment 1 of 3

10     while p<20000000
20        p=nxtprm(p)
30        inc n
40        if sum @ p = 0 then print p,n,sum,sum//p
50        sum=sum+p
60     wend

finds

`   prime        n  sum of prev  multiple      2        1            0       0      5        3            5       1     71       20          568       8 369119    31464   5536785000   15000`

So the two primes of the first part are 5 and 71, as, presumably, 2 doesn't count even though it "divides" zero evenly.

For the second part, M is 369,119 and n is 31,464. The sum of the first 31,463 primes is 5,536,785,000, which is 15,000 times 369,119.

BTW the program checks up to 20,000,000 and doesn't find another case.

 Posted by Charlie on 2012-02-12 18:26:57

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