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A113635
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Sum of the first 7^n primes.
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0
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2, 58, 4888, 363288, 24047406, 1482656786, 87401659166, 4997438572618, 279544493456056, 15382405126365576, 835737977869494888, 44947274043643171988
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OFFSET
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0,1
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COMMENTS
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We can easily show that the sum of the first k^n primes is an even number for odd k. However except for the first term the last digit of these numbers is 6 or 8. What happened to 0 and 4? Similarly, there is no 4 ending for the known values of sums of the first 5^n primes. This is just an example of the Strong law of small numbers (R Guy). Indeed, The sum of the occurrences of any digit 0-9 in the sum of the first n prime numbers, A007504(n),-> Pi(n)/10 as n -> infinity. For n = 10^6, Pi(n) = 78498 and the count of the last digit d in A007504(10^6) = d = 0 1 2 3 4 5 6 7 8 9 count = 7919,7963,7808,7840,8000,7759,7778,7878,7744,7809 While we probably will never know, a(n) will end in all even numbers in similar fashion as n-> infinity. Any guesses? Maybe there is an analytical way to prove this? a(12) will be forthcomming in a week or so.
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LINKS
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Table of n, a(n) for n=0..11.
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EXAMPLE
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The first 7^1 primes add up to 58
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MATHEMATICA
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t = {}; c = 1; k = 3; s = 2; Do[While[c < 7^n, If[ PrimeQ@k, c++; s += k]; k += 2]; Print@s; AppendTo[t, s], {n, 0, 9}]; t (from Robert G. Wilson v, Jan 17 2006)
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CROSSREFS
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Cf. A007504.
Sequence in context: A115610 A191798 A206531 * A136095 A181866 A106897
Adjacent sequences: A113632 A113633 A113634 * A113636 A113637 A113638
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KEYWORD
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hard,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Jan 15 2006
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STATUS
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approved
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