A029479 Numbers k that divide the (left) concatenation of all numbers <= k written in base 10 (most significant digit on left).

1, 3, 9, 19, 27, 41, 103, 147, 189, 441, 567, 711, 6759, 15353, 24441, 59823, 209903, 1430217, 2848851, 2969973, 13358067, 146247471, 289542573, 1891846557, 2388085659, 4489093899, 5345125899, 5455876131, 9843149241

Offset

1,2

Comments

No other terms below 10^10.

Links

Table of n, a(n) for n=1..29.

Index entries for related sequences

Example

19 is a term since 19181716151413121110987654321 is divisible by 19.

Mathematica

b = 10; c = {}; Select[Range[10^4], Divisible[FromDigits[c = Join[IntegerDigits[#, b], c], b], #] &] (* Robert Price, Mar 12 2020 *)
Select[Range[134*10^5], Divisible[FromDigits[Flatten[IntegerDigits/@Range[#, 1, -1]]], #]&] (* Harvey P. Dale, Oct 09 2022 *)

Prog

(Python)
def concat_mod(base, k, mod):
  total, offset, digits, n1 = 0, 0, 1, 1
  while n1 <= k:
    n2, p = min(n1*base-1, k), n1*base
    # Compute ((p-1)*n2-1)*p**(n2-n1+1)-(n1-1)*p+n1 divided by (p-1)**2.
    # Since (a//b)%mod == (a%(b*mod))//b, compute the numerator mod (p-1)**2*mod.
    tmp = pow(p, n2-n1+1, (p-1)**2*mod)
    tmp = ((p-1)*n2-1)*tmp-(n1-1)*p+n1
    tmp = (tmp%((p-1)**2*mod))//(p-1)**2
    total += tmp*pow(base, offset, mod)
    offset, digits, n1 = offset+digits*(n2-n1+1), digits+1, p
  return total%mod
for k in range(1, 10**10):
  if concat_mod(10, k, k) == 0: print(k) # Jason Yuen, Jan 14 2024

Crossrefs

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Sequence in context: A069832 A057257 A029464 * A215803 A014869 A018495

Adjacent sequences: A029476 A029477 A029478 * A029480 A029481 A029482

Keyword

nonn,base,more

Author

Olivier Gérard

Extensions

6759 from Andrew Gacek (andrew(AT)dgi.net), Feb 20 2000

More terms from Larry Reeves (larryr(AT)acm.org), May 24 2001

Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12 2002

a(18)-a(21) from Max Alekseyev, May 15 2011

a(22)-a(29) from Jason Yuen, Jan 14 2024

Status

approved