0,1

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.

The first 7^1 primes add up to 58

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 (rgwv(at)rgwv.com), Jan 17 2006)

Cf. A007504.

Sequence in context: A121931 A156507 A115610 this_sequence A136095 A106897 A141869

Adjacent sequences: A113632 A113633 A113634 this_sequence A113636 A113637 A113638

hard,nonn

Cino Hilliard (hillcino368(AT)gmail.com), Jan 15 2006