A300517 -id:A300517 - cp's OEIS frontend

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

David Cleaver, Mar 23 2023

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

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

Cf. A055642, A098129, A300517.

Partial sums of A110803.

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

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))

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

# 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

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

A098129 Concatenate 1 once, 2 twice, 3 three times, up to n n times.

Original entry on oeis.org

1, 122, 122333, 1223334444, 122333444455555, 122333444455555666666, 1223334444555556666667777777, 122333444455555666666777777788888888, 122333444455555666666777777788888888999999999

Offset: 1

Michael Joseph Halm, Jan 04 2005

a(n) is composite for all 2 <= n <= 1000. - David Cleaver, Mar 22 2023

			a(4) = 1223334444 because 1 concatenated once then concatenated with 2 twice and 3 three times and 4 four times gives 1223334444.

Seiichi Manyama, Table of n, a(n) for n = 1..31

Cf. A000461, A300517, A361751 (number of decimal digits).

a:= n-> parse(cat(seq(i$i, i=1..n))):
seq(a(n), n=1..12);  # Alois P. Heinz, Mar 07 2018

nn = 12; a[0] = {}; Do[Set[a[n], Join[a[n - 1], Flatten@ ConstantArray[IntegerDigits[n], n]]], {n, nn}]; Array[FromDigits @* a, nn] (* Michael De Vlieger, Mar 29 2023 *)

a(n) = {my(a=0,i,k);
  for(i=1,n, k = logint(i,10)+1;
    a = a*10^(i*k) + i*(10^(i*k)-1)/(10^k-1);
); return(a); } \\ David Cleaver, Mar 29 2023

def A098129(n): return int(''.join(str(j)*j for j in range(1,n+1))) # Chai Wah Wu, Mar 29 2023

Offset and a(8) corrected by Seiichi Manyama, Mar 07 2018

A300558 a(n) is the concatenation n written in base 2 n times, n-1 written in base 2 n-1 times, ..., 1 written in base 2 once.

Original entry on oeis.org

1, 10101, 11111110101, 10010010010011111110101, 10110110110110110010010010011111110101, 11011011011011011010110110110110110010010010011111110101, 11111111111111111111111011011011011011010110110110110110010010010011111110101

Offset: 1

Seiichi Manyama, Mar 08 2018

Seiichi Manyama, Table of n, a(n) for n = 1..21

Cf. A007088, A300517, A300557.

def A300558(n)
  a = '1'
  [1] + (2..n).map{|i| a = (i.to_s(2) * i + a.to_s).to_i}
end
p A300558(10)

cp's OEIS Frontend

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

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A098129 Concatenate 1 once, 2 twice, 3 three times, up to n n times.

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A300558 a(n) is the concatenation n written in base 2 n times, n-1 written in base 2 n-1 times, ..., 1 written in base 2 once.

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Ruby