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%I #19 Jan 26 2019 19:35:14
%S 1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,
%T 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,
%U 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N Number of digits in sum of first n primes (A007504).
%C Since A007504(n) has the asymptotic expression ~ n^2 * log(n) / 2, a(n) has the asymptotic expression n^2 * log(n) / 2 = floor(log_10(10* n^2 * log(n) / 2)) = floor(log_10(5* n^2 * log(n))) = floor(log_10(5) + log_10(n^2) + log_10(log(n))) = floor(0.698970004 + 2*log_10(n) + log_10(log(n))). What is the smallest n such that a(n) = 5, 6, 7, ...?
%H Harvey P. Dale, <a href="/A123119/b123119.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A055642(A007504(n)) = floor(log_10(10*A007504(n))) = A004216(A007504(n)) + 1 = A004218(A007504(n) + 1).
%e a(3) = 2 because 2 + 3 + 5 = 10 has 2 digits in its decimal expansion.
%t f[n_] := Floor[ Log[10, Sum[Prime@i, {i, n}]] + 1]; Array[f, 105] (* _Robert G. Wilson v_ *)
%t f[n_] := IntegerLength[Total[Prime[Range[n]]]]; Array[f, 105] (* _Jan Mangaldan_, Jan 04 2017 *)
%t IntegerLength/@Accumulate[Prime[Range[110]]] (* _Harvey P. Dale_, Jan 26 2019 *)
%Y Cf. A000040, A000041, A004216, A004218, A034386, A055642, A111287.
%K base,nonn
%O 1,3
%A _Jonathan Vos Post_, Sep 28 2006
%E More terms from _Robert G. Wilson v_, Oct 05 2006
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