A068670 Number of digits in the concatenation of first n primes.

0, 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154

Offset: 0

Author

Eugene McDonnell (eemcd(AT)mac.com), Jan 18 2004

Keywords

• nonn
• base

Comments

Partial sums of A097944. - Lekraj Beedassy, Aug 26 2008

Examples

			a(5) is 6 because concatenating the first five primes gives 235711, which has six digits.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10000. (Initial 0 added by _M. F. Hasler_, Nov 02 2019.)
• Eric Weisstein's World of Mathematics, Copeland-Erdos Constant

Crossrefs

Cf. A019518, A097944 (number of decimal digits of the primes).

Cf. A033308 (decimal expansion of the Copeland-Erdos constant).

Programs

Magma

a068670:=func< n | n + &+[ Floor(Log(10, NthPrime(k))): k in [1..n] ] >; [ a068670(n): n in [1..70] ];

Mathematica

Table[n + Sum[Floor[Log[10, Prime[k]]], {k, 1, n}], {n, 1, 90}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006 *)
Accumulate[IntegerLength[Prime[Range[70]]]] (* Harvey P. Dale, Jul 01 2012 *)

PARI

A68670=List(0); A068670(n)={for(N=#A68670,n, listput(A68670, A68670[N] + A097944(N))); A68670[n+1]} \\ M. F. Hasler, Oct 24 2019

Python

from sympy import sieve
from itertools import accumulate, chain
def f(, n): return  + len(str(n))
def agen(): yield from accumulate(chain((0,), (p for p in sieve)), f)
print(list(islice(agen(), 62))) # Michael S. Branicky, Feb 03 2023

Formula

a(n) = Sum_{i=1..n} ceiling(log_10(1 + prime(i))). - Daniel Forgues, Apr 02 2014

Extensions

Extended to a(0) = 0 by M. F. Hasler, Oct 24 2019