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

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

1,2

Comments

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.

Links

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

Example

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

Maple

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:

Prog

(Python)
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

Crossrefs

Cf. A094151, A110740.

Sequence in context: A036404 A347355 A186077 * A018498 A002048 A174989

Adjacent sequences: A358607 A358608 A358609 * A358611 A358612 A358613

Keyword

nonn,base

Author

_Martin Renner_, Nov 23 2022

Status

approved