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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.
4
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
1,1
COMMENTS
a(n) = the smallest number k such that k*n is a number from A276349.
a(n) > 0 for all n.
REFERENCES
L. Pick, Dirichletovy šuplíčky. Pokroky matematiky, fyziky & astronomie; 2 (2016), 106-118. (In Czech; The Dirichlet pigeonhole principle)
LINKS
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)
EXAMPLE
For n=3; 3*370 = 1110 (term of A276349).
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 *)
PROG
(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;
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Jaroslav Krizek, Aug 30 2016
STATUS
approved