A098129 -id:A098129 - 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

A358275 Least prime factor of A098129(n).

Original entry on oeis.org

2, 71, 2, 5, 2, 1141871, 2, 3, 2, 58728589, 2, 3, 2, 5, 2, 3, 2, 277, 2, 4643, 2, 29, 2, 5, 2, 3, 2, 37, 2, 3, 2, 13, 2, 3, 2, 264439098646852541, 2, 7, 2, 53, 2, 7, 2, 3, 2, 587, 2, 3, 2, 45307, 2, 3, 2, 5, 2, 11, 2, 7, 2, 13, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 983, 2, 5, 2, 53, 2, 11

Offset: 2

David Cleaver, Mar 26 2023

nonn

			a(3) = 71 because 71 is the smallest prime factor of A098129(3) = 122333.
a(7) = 1141871 because 1141871 is the smallest prime factor of A098129(7) = 1223334444555556666667777777.

Cf. A020639, A098129.

from sympy import primefactors
def A358275(n): return min(primefactors(int(''.join(str(j)*j for j in range(1,n+1))))) if n&1 else 2 # Chai Wah Wu, Apr 15 2023

a(n) = A020639(A098129(n)).

a(n) = 2 if n is even. - Chai Wah Wu, Apr 15 2023

A300517 a(n) is the concatenation of n n times, n-1 n-1 times, ..., 22, 1.

Original entry on oeis.org

1, 221, 333221, 4444333221, 555554444333221, 666666555554444333221, 7777777666666555554444333221, 888888887777777666666555554444333221, 999999999888888887777777666666555554444333221, 10101010101010101010999999999888888887777777666666555554444333221

Offset: 1

Seiichi Manyama, Mar 07 2018

a(n) is composite for all 2 <= n <= 1000. - David Cleaver, Apr 14 2023

			a(4) = 4444333221 because 4 concatenated four times then concatenated with 3 three times and 2 twice and 1 once gives 4444333221.

Alois P. Heinz, Table of n, a(n) for n = 1..31

Cf. A000461, A098129, A361751 (number of decimal digits).

a:= n-> parse(cat(seq((n-i)$(n-i), i=0..n-1))):
seq(a(n), n=1..12);  # Alois P. Heinz, Mar 07 2018

a(n) = {my(a=0, b=0, d=1, i);
  for(i=1, n, b = logint(i, 10)+1;
    a += d*i*(10^(i*b)-1)/(10^b-1);
    d *= 10^(i*b); ); return(a); } \\ David Cleaver, Apr 14 2023

def A300517(n)
  a = '1'
  [1] + (2..n).map{|i| a = (i.to_s * i + a.to_s).to_i}
end
p A300517(20)

A300557 a(n) is the concatenation 1 written in base 2 once, 2 written in base 2 twice, ..., n written in base 2 n times.

Original entry on oeis.org

1, 11010, 11010111111, 11010111111100100100100, 11010111111100100100100101101101101101, 11010111111100100100100101101101101101110110110110110110, 11010111111100100100100101101101101101110110110110110110111111111111111111111

Offset: 1

Seiichi Manyama, Mar 08 2018

nonn

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

Cf. A007088, A098129, A300558.

A105310 a(n) = concatenation of i n times, with i = 1 to n.

Original entry on oeis.org

1, 1122, 111222333, 1111222233334444, 1111122222333334444455555, 111111222222333333444444555555666666, 1111111222222233333334444444555555566666667777777

Offset: 21

Alexandre Wajnberg, Apr 25 2005

			a(10)=concatenation of (1^10)(2^10)(3^10)...(9^10)(10^10)

F. Smarandache, "Properties of the numbers", Univ. of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.

Eric Weisstein's World of Mathematics, Smarandache Sequences.

Cf. A098129, A000461.

A105311 a(n) = n concatenations of numbers from 1 to n, concatenated.

Original entry on oeis.org

1, 1212, 123123123, 1234123412341234, 1234512345123451234512345, 123456123456123456123456123456123456, 1234567123456712345671234567123456712345671234567, 1234567812345678123456781234567812345678123456781234567812345678, 123456789123456789123456789123456789123456789123456789123456789123456789123456789

Offset: 1

Alexandre Wajnberg, Apr 25 2005

F. Smarandache, "Properties of the numbers", Univ. of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.

Eric Weisstein's World of Mathematics, Smarandache Sequences.

Cf. A098129, A000461.

a(5) corrected and more terms from Georg Fischer, Nov 13 2024

A323426 a(n) = decimal concatenation of n (once), n-1 (twice), n-2 (3 times), ..., 1 (n times).

Original entry on oeis.org

1, 211, 322111, 4332221111, 544333222211111, 655444333322222111111, 7665554444333332222221111111, 877666555544444333333222222211111111

Offset: 1

Eder Vanzei, Aug 30 2019

a(2), a(3) and a(4) are primes.

Cf. A098129.

cp's OEIS Frontend

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

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A358275 Least prime factor of A098129(n).

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A300517 a(n) is the concatenation of n n times, n-1 n-1 times, ..., 22, 1.

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A300557 a(n) is the concatenation 1 written in base 2 once, 2 written in base 2 twice, ..., n written in base 2 n times.

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A105310 a(n) = concatenation of i n times, with i = 1 to n.

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A105311 a(n) = n concatenations of numbers from 1 to n, concatenated.

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A323426 a(n) = decimal concatenation of n (once), n-1 (twice), n-2 (3 times), ..., 1 (n times).

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