A029479 - OEIS

%I #32 Jan 22 2024 06:06:27

%S 1,3,9,19,27,41,103,147,189,441,567,711,6759,15353,24441,59823,209903,

%T 1430217,2848851,2969973,13358067,146247471,289542573,1891846557,

%U 2388085659,4489093899,5345125899,5455876131,9843149241

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

%C No other terms below 10^10.

%H <a href="/index/N#concat">Index entries for related sequences</a>

%e 19 is a term since 19181716151413121110987654321 is divisible by 19.

%t b = 10; c = {}; Select[Range[10^4], Divisible[FromDigits[c = Join[IntegerDigits[#, b], c], b], #] &] (* _Robert Price_, Mar 12 2020 *)

%t Select[Range[134*10^5],Divisible[FromDigits[Flatten[IntegerDigits/@Range[#,1,-1]]],#]&] (* _Harvey P. Dale_, Oct 09 2022 *)

%o (Python)

%o def concat_mod(base, k, mod):

%o   total, offset, digits, n1 = 0, 0, 1, 1

%o   while n1 <= k:

%o     n2, p = min(n1*base-1, k), n1*base

%o     # Compute ((p-1)*n2-1)*p**(n2-n1+1)-(n1-1)*p+n1 divided by (p-1)**2.

%o     # Since (a//b)%mod == (a%(b*mod))//b, compute the numerator mod (p-1)**2*mod.

%o     tmp = pow(p,n2-n1+1,(p-1)**2*mod)

%o     tmp = ((p-1)*n2-1)*tmp-(n1-1)*p+n1

%o     tmp = (tmp%((p-1)**2*mod))//(p-1)**2

%o     total += tmp*pow(base,offset,mod)

%o     offset, digits, n1 = offset+digits*(n2-n1+1), digits+1, p

%o   return total%mod

%o for k in range(1,10**10):

%o   if concat_mod(10, k, k) == 0: print(k) # _Jason Yuen_, Jan 14 2024

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

%K nonn,base,more

%O 1,2

%A _Olivier Gérard_

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

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

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

%E a(18)-a(21) from _Max Alekseyev_, May 15 2011

%E a(22)-a(29) from _Jason Yuen_, Jan 14 2024