A029455 Numbers k that divide the (right) concatenation of all numbers <= k written in base 10 (most significant digit on left).

1, 2, 3, 5, 6, 9, 10, 12, 15, 18, 20, 25, 27, 30, 36, 45, 50, 54, 60, 69, 75, 90, 100, 108, 120, 125, 135, 150, 162, 180, 200, 216, 225, 248, 250, 270, 300, 324, 360, 375, 405, 450, 470, 500, 540, 558, 600, 648, 675, 710, 750, 810, 900, 1000, 1053, 1080, 1116

Offset

1,2

Comments

Numbers k such that k divides A007908(k).

Links

Jason Yuen, Table of n, a(n) for n = 1..1195 (terms 1..236 from M. F. Hasler, terms 237..637 from Chai Wah Wu)

Example

k = 13 is not a term since 12345678910111213 is not divisible by 13.

Mathematica

b = 10; c = {}; Select[Range[10^5], Divisible[FromDigits[c = Join[c, IntegerDigits[#, b]], b], #] &] (* Robert Price, Mar 11 2020 *)
Select[Range[1200], Divisible[FromDigits[Flatten[IntegerDigits/@Range[#]]], #]&] (* Harvey P. Dale, Dec 31 2020 *)
nxt[{rc_, n_}]:={rc*10^IntegerLength[n+1]+n+1, n+1}; Select[NestList[nxt, {1, 1}, 1200], Mod[#[[1]], #[[2]]]==0&][[;; , 2]] (* Harvey P. Dale, Sep 26 2023 *)

Prog

(PARI) c=0; for(d=1, 1e9, for(n=d, -1+d*=10, (c=c*d+n)%n || print1(n", ")); d--) \\ M. F. Hasler, Sep 11 2011
(Python)
A029455_list, r = [], 0
for n in range(1, 10**4+1):
    r = r*10**len(str(n))+n
    if not (r % n):
        A029455_list.append(n) # Chai Wah Wu, Nov 05 2014
(Python)
def concat_mod(base, k, mod):
  total, digits, n1 = 0, 1, 1
  while n1 <= k:
    n2, p = min(n1*base-1, k), n1*base
    # Compute ((p-1)*n1+1)*p**(n2-n1+1)-(n2+1)*p+n2 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)*n1+1)*tmp-(n2+1)*p+n2
    tmp = (tmp%((p-1)**2*mod))//(p-1)**2
    total = (total*pow(p, n2-n1+1, mod)+tmp)%mod
    digits, n1 = digits+1, p
  return total
for k in range(1, 10**10+1):
  if concat_mod(10, k, k) == 0: print(k) # Jason Yuen, Jan 27 2024

Crossrefs

Cf. A007908.

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978, A332580.

See A171785 for numbers that divide the concatenation of a(1) through a(n).

Sequence in context: A288480 A344633 A029451 * A234718 A341158 A294849

Adjacent sequences: A029452 A029453 A029454 * A029456 A029457 A029458

Keyword

nonn,base

Author

Olivier Gérard

Status

approved