A094151 -id:A094151 - cp's OEIS frontend

A110740 Numbers k such that the concatenation 1,2,3,...,(k-1) is divisible by k.

1, 3, 9, 27, 69, 1053, 1599, 2511, 8167, 21371, 73323, 225681, 313401, 362703, 371321, 1896939, 2735667, 3426273, 3795093, 5433153, 302278903, 1371292077, 19755637749, 23560349643, 33184178631

Offset: 1

Amarnath Murthy, Aug 10 2005

Subsequence of A029455 composed of the terms coprime to 10. - Max Alekseyev, Jun 07 2023

a(26) > 10^11. - Jason Yuen, Oct 12 2024

			3 divides 12, 9 divides 12345678.

Cf. A007908, A029455, A094151, A171785, A332580, A362966.

s = ""; Do[s = s <> ToString[n]; If[Mod[ToExpression[s], n + 1] == 0, Print[n + 1]], {n, 0, 5*10^6}] (* Ryan Propper, Aug 28 2005 *)
Select[Range[55*10^5],Divisible[FromDigits[Flatten[IntegerDigits/@Range[ #-1]]],#]&] (* Harvey P. Dale, Mar 28 2020 *)

# See A029455 for concat_mod
def isok(k): return concat_mod(10, k-1, k)==0 # Jason Yuen, Oct 06 2024

More terms from Ryan Propper, Aug 28 2005
a(20) from Giovanni Resta, Apr 10 2018
a(21)-a(24) from Scott R. Shannon, using a modified version of an algorithm by Joseph Myers, Apr 10 2020
a(25) from Jason Yuen, Oct 06 2024

A095221 (Concatenation 1,2,3,...,n) mod n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 5, 6, 0, 0, 4, 0, 4, 4, 0, 12, 15, 0, 16, 0, 3, 4, 6, 12, 0, 16, 0, 8, 3, 0, 18, 28, 15, 16, 25, 0, 14, 10, 33, 20, 27, 24, 27, 4, 0, 36, 35, 12, 29, 0, 48, 4, 14, 0, 15, 4, 18, 28, 3, 0, 28, 60, 45, 60, 55, 48, 13, 52, 0, 50, 14, 36, 12, 72, 0, 20, 4, 72, 74, 60, 54, 24, 65

Offset: 1

Amarnath Murthy, Jun 10 2004

			a(4) = 1234 mod 4 = 2.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A007908, A094151.

from itertools import count, islice
def agen():
    s = "0"
    for n in count(1): s += str(n); yield int(s)%n
print(list(islice(agen(), 83))) # Michael S. Branicky, Nov 21 2022

a(n) = A007908(n) mod n. - Michel Marcus, Nov 21 2022

More terms from John W. Layman, Jul 29 2005

A358610 Numbers k such that the concatenation 1,2,3,... up to (k-1) is one less than a multiple of k.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 20, 25, 40, 50, 52, 100, 125, 200, 250, 400, 475, 500, 601, 848, 908, 1000, 1120, 1250, 1750, 2000, 2500, 2800, 2900, 3670, 4000, 4375, 4685, 5000, 6085, 7000, 7640, 7924, 8375, 10000, 10900, 12500, 13346, 14000, 17800, 20000, 21568, 25000

Offset: 1

Martin Renner, Nov 23 2022

For a >= 0, the infinite subsequence of numbers 10^a, 2^b*10^a (for 1 <= b <= 2) and 5^c*10^a (for 1 <= c <= 3), i.e., 1, 2, 4, 5, 10, 20, 25, 40, 50, 100, 125, 200, 250, 400, 500, 1000, 1250, 2000, 2500, 4000, 5000, ... are terms in the sequence because first, the concatenation 1, 2, 3, ... up to (10^a - 1) mod 10^a is equal to 10^a times the concatenation 1, 2, 3, ... up to (10^a - 2) + (10^a - 1) mod 10^a, which results in 10^a - 1 and second, the concatenation 1, 2, 3, ... up to (2^b*10^a - 1) mod 2^b*10^a is equal to 10^(a+1) times the concatenation 1, 2, 3, ... up to (2^b*10^a - 2) + (2^b*10^a - 1) mod 2^b*10^a, which results in 2^b*10^a - 1 and third, the concatenation 1, 2, 3, ... up to (5^c*10^a - 1) mod 5^c*10^a is equal to 10^(a+c) times the concatenation 1, 2, 3, ... up to (5^c*10^a - 2) + (5^c*10^a - 1) mod 5^c*10^a, which results in 5^c*10^a - 1.

			13 is a term because 123456789101112 mod 13 = 12.
20 is a term because 12345678910111213141516171819 mod 20 = 19.

Martin Renner, Table of n, a(n) for n = 1..92

Cf. A094151, A110740.

a:=proc(m)
  local A, str, i;
  if m = 1 then return([1]);
  else
    if m = 2 then return([1, 2]);
    else
      A := [1, 2];
      str := 1;
      for i from 2 to m do
        str := str*10^length(i) + i;
        if str mod (i+1) = i then A := [op(A), i+1]; fi;
      od;
    fi;
  fi;
  return(A);
end:

from itertools import count, islice
def agen():
    s = "0"
    for n in count(1):
        if int(s)%n == n - 1: yield n
        s += str(n)
print(list(islice(agen(), 30))) # Michael S. Branicky, Nov 23 2022

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A110740 Numbers k such that the concatenation 1,2,3,...,(k-1) is divisible by k.

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A095221 (Concatenation 1,2,3,...,n) mod n.

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A358610 Numbers k such that the concatenation 1,2,3,... up to (k-1) is one less than a multiple of k.

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