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A061963 Numbers n such that n divides the (left) concatenation of all numbers <= n written in base 10 (most significant digit on right).

%I A061963 #32 Jan 14 2025 17:59:04
%S A061963 1,3,9,189,753,987,6739,10953,51963,171897,224081,635031,1135001,
%T A061963 4437459
%N A061963 Numbers n such that n divides the (left) concatenation of all numbers <= n written in base 10 (most significant digit on right).
%C A061963 This sequence differs from A029527 in that all least significant zeros are kept during concatenation.
%C A061963 Left concatenation, reverse order (i.e., digit-wise reversal of the concatenation 123...n), as in A138793.
%C A061963 No more terms < 10^7.
%C A061963 All terms must be odd.
%H A061963 <a href="/index/N#concat">Index entries for related sequences</a>
%e A061963 n = 13 is not a term since 31211101987654321 is not divisible by 13. (Note that the order of the digits of 13, 12 and 10 is reversed.)
%e A061963 See A061955 for further examples.
%t A061963 k = 2; lst = {}; rid = 1; While[k < 1001, exp = Floor[ Log10[rid]] + 1 + If[Mod[k, 10] == 1, IntegerExponent[k - 1, 10], 0]; rid = rid + FromDigits@ Reverse@ IntegerDigits@ k*(10^exp); If[ Mod[rid, k] == 0, Print@ k; AppendTo[lst,k]]; k++]; lst (* and to test any single value n *) fQ[n_] := Mod[ FromDigits@ Reverse@ Flatten@ IntegerDigits@ Range@ n, n] == 0 (* _Robert G. Wilson v_, Sep 12 2011 *)
%t A061963 Select[Range[5*10^6],Divisible[FromDigits[Reverse[Flatten[ IntegerDigits/@ Range[ #]]]], #]&] (* _Harvey P. Dale_, Apr 10 2017 *)
%o A061963 (PARI) isok(n) = my(s = ""); forstep (k=n, 1, -1, sk = digits(k); forstep (j=#sk, 1, -1, s = concat(s, sk[j]))); (eval(s) % n) == 0; \\ _Michel Marcus_, Jan 28 2017
%Y A061963 Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.
%Y A061963 Cf. A138793.
%K A061963 nonn,base,more
%O A061963 1,2
%A A061963 Larry Reeves (larryr(AT)acm.org), May 24 2001
%E A061963 Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12 2002
%E A061963 a(12)-a(14) from _Lars Blomberg_, Aug 19 2011