A123119 Number of digits in sum of first n primes (A007504).

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, 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, 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

Offset: 1

Author

Jonathan Vos Post, Sep 28 2006

Keywords

• base
• nonn

Comments

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, ...?

Examples

			a(3) = 2 because 2 + 3 + 5 = 10 has 2 digits in its decimal expansion.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A000041, A004216, A004218, A034386, A055642, A111287.

Programs

Mathematica

f[n_] := Floor[ Log[10, Sum[Prime@i, {i, n}]] + 1]; Array[f, 105] (* Robert G. Wilson v *)
f[n_] := IntegerLength[Total[Prime[Range[n]]]]; Array[f, 105] (* Jan Mangaldan, Jan 04 2017 *)
IntegerLength/@Accumulate[Prime[Range[110]]] (* Harvey P. Dale, Jan 26 2019 *)

Formula

a(n) = A055642(A007504(n)) = floor(log_10(10*A007504(n))) = A004216(A007504(n)) + 1 = A004218(A007504(n) + 1).

Extensions

More terms from Robert G. Wilson v, Oct 05 2006