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

1,3

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, ...?

Links

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

Formula

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A366721 A080342 A081604 * A099396 A126235 A220104

Adjacent sequences: A123116 A123117 A123118 * A123120 A123121 A123122

Keyword

base,nonn

Author

Jonathan Vos Post, Sep 28 2006

Extensions

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

Status

approved