A276348 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.

10, 5, 370, 25, 2, 185, 158730, 125, 123456790, 1, 10, 925, 85470, 79365, 74, 625, 653594771241830, 61728395, 58479532163742690, 5, 52910, 5, 483091787439613526570, 4625, 4, 42735, 41152263374485596707818930, 396825, 383141762452107279693486590, 37

Offset: 1

Author

Jaroslav Krizek, Aug 30 2016

Keywords

• nonn
• base

Comments

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

a(n) > 0 for all n.

Examples

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

References

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

Links

• Robert Israel, Table of n, a(n) for n = 1..1016

Crossrefs

Cf. A052983, A084680, A276349.

Programs

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;

Maple

f:= proc(n) local b,c,d,m,q;
    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);
     2^(d-b)*5^(d-c)*(10^q-1)/(9*m)
end proc:
map(f, [$1..100]); # Robert Israel, Aug 30 2016

Mathematica

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 *)

Formula

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

From Robert Israel, Aug 30 2016: (Start)

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

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

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