A276348 - OEIS

%I #16 Sep 08 2022 08:46:17

%S 10,5,370,25,2,185,158730,125,123456790,1,10,925,85470,79365,74,625,

%T 653594771241830,61728395,58479532163742690,5,52910,5,

%U 483091787439613526570,4625,4,42735,41152263374485596707818930,396825,383141762452107279693486590,37

%N a(n) = the smallest number k such that k*n is a number with a string of 1's followed by a string of 0's.

%C a(n) = the smallest number k such that k*n is a number from A276349.

%C a(n) > 0 for all n.

%D L. Pick, Dirichletovy šuplíčky. Pokroky matematiky, fyziky & astronomie; 2 (2016), 106-118. (In Czech; The Dirichlet pigeonhole principle)

%H Robert Israel, <a href="/A276348/b276348.txt">Table of n, a(n) for n = 1..1016</a>

%F a(n) = A052983(n)/n.

%F From _Robert Israel_, Aug 30 2016: (Start)

%F Let n = 2^b*5^c*m where GCD(m,10)=1, and q = A084680(9*m).

%F If b=c=0 let d=1, otherwise d=max(b,c).

%F Then a(n) = 2^(d-a)*5^(d-b)*(10^q-1)/(9*m). (End)

%e For n=3; 3*370 = 1110 (term of A276349).

%p f:= proc(n) local b,c,d,m,q;

%p     b:= padic:-ordp(n,2); c:= padic:-ordp(n,5); if b+c=0 then d:= 1 else d:= max(b,c) fi; m:= n/2^b/5^c; q:= numtheory:-order(10,9*m);

%p      2^(d-b)*5^(d-c)*(10^q-1)/(9*m)

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Aug 30 2016

%t Table[k = 1; While[! If[Length@ # == 2, Flatten@ Map[Union, #] == {1, 0}, False] &@ Split@ IntegerDigits[k n], k++]; k, {n, 8}] (* _Michael De Vlieger_, Aug 30 2016 *)

%o (Magma) a:=10; S:=[a]; for n in [2..6] do k:=0; flag:= true; while flag do k+:=1; if [k*n] subset [n: n in [1..10000] | Seqint(Setseq(Set(Sort(Intseq(n))))) eq 10 and Seqint(Sort((Intseq(n)))) eq n] then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;

%Y Cf. A052983, A084680, A276349.

%K nonn,base

%O 1,1

%A _Jaroslav Krizek_, Aug 30 2016