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

1,2

Links

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

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

Example

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

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

Prog

(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

Crossrefs

Cf. A055642, A098129, A300517.

Partial sums of A110803.

Sequence in context: A165145 A046489 A277209 * A089717 A050760 A075057

Adjacent sequences: A361748 A361749 A361750 * A361752 A361753 A361754

Keyword

nonn,base,easy

Author

David Cleaver, Mar 23 2023

Status

approved