A361751 a(n) is the number of decimal digits in A098129(n) and A300517(n).

1, 3, 6, 10, 15, 21, 28, 36, 45, 65, 87, 111, 137, 165, 195, 227, 261, 297, 335, 375, 417, 461, 507, 555, 605, 657, 711, 767, 825, 885, 947, 1011, 1077, 1145, 1215, 1287, 1361, 1437, 1515, 1595, 1677, 1761, 1847, 1935, 2025, 2117, 2211, 2307, 2405, 2505, 2607, 2711, 2817, 2925

Offset: 1

Author

David Cleaver, Mar 23 2023

Keywords

• nonn
• base
• easy

Examples

			For n = 4, a(4) = 10, because A098129(4) = 1223334444.
For n = 10, a(10) = 65, because A098129(10) = 12233344445555566666677777778888888899999999910101010101010101010.
		

Links

• Winston de Greef, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A055642, A098129, A300517.

Partial sums of A110803.

Programs

Maple

a:= proc(n) a(n):= `if`(n<1, 0, a(n-1)+n*length(n)) end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 23 2023

PARI

a(n) = {my(x=logint(n,10)+1);x*n*(n+1)/2 - ((100^x-1)/99 - (10^x-1)/9)/2}
vector(100, i, a(i))

Python

def a(n):
    d = len(str(n))
    m = 10**d
    return d*n*(n+1)//2 - ((m-11)*m + 10)//198
print([a(n) for n in range(1, 55)]) # Michael S. Branicky, Mar 24 2023 modified Mar 29 2023

Python

# faster for generating initial segment of sequence
from itertools import count, islice
def agen(s=0): yield from (s:=s+n*len(str(n)) for n in count(1))
print(list(islice(agen(), 60))) # Michael S. Branicky, Mar 24 2023

Formula

a(n) = A055642(A098129(n)).

From Alois P. Heinz, Mar 23 2023: (Start)

a(n) = Sum_{j=1..n} j*A055642(j).

a(n) = Sum_{j=1..n} A110803(j). (End)

a(n) = Sum_{k=0..floor(log_10(n))} (n*(n+1) - 10^k*(10^k-1))/2. - Andrew Howroyd, Mar 24 2023

a(n) = k*n*(n+1)/2 - ((100^k-1)/99 - (10^k-1)/9)/2, where k = floor(log_10(n))+1. - David Cleaver, Mar 25 2023