A187041 Numbers for which Midy's theorem does not hold.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 30, 31, 32, 33, 36, 37, 39, 40, 41, 42, 43, 45, 48, 50, 51, 53, 54, 57, 60, 62, 63, 64, 66, 67, 69, 71, 72, 74, 75, 78, 79, 80, 81, 82, 83, 84, 86, 87, 90, 93, 96, 99, 100, 102, 105, 106, 107, 108, 111, 114, 117, 119, 120, 123, 124, 125, 126, 128, 129, 132, 134, 135, 138, 141, 142, 144, 147, 148, 150

Offset: 1

Author

Jani Melik, Mar 02 2011

Keywords

• nonn
• base

Links

• Wikipedia, Midy's theorem

Crossrefs

Cf. A028416, A187040.

Programs

Maple

fct1 := proc(an) local i,st:  st := 0:
for i from 1 to nops(an)/2 do
   st := op(i,an)*10^(nops(an)/2-i) + st
od: RETURN(st):  end:
fct2 := proc(an) local i,st:  st := 0:
for i from nops(an)/2+1 to nops(an) do
   st := op(i,an)*10^(nops(an)/2-i+nops(an)/2) + st
od:  RETURN(st):  end:
A187041 := proc(n) local st:
st := op(4,numtheory[pdexpand](1/n));
if (modp(nops(st),2) <> 0 or nops(st) = 1 or n = 1) then
     RETURN(n)
elif (modp(nops(st),2) = 0) then
   if not(10^(nops(st)/2)-1 - (fct1(st)+fct2(st)) = 0) then
       RETURN(n)
fi: fi: end:  seq(A187041(n), n=1..250);

Mathematica

okQ[n_] := Module[{ps = First /@ FactorInteger[n], d, len}, If[n < 2 || Complement[ps, {2, 5}] == {}, False, d = RealDigits[1/n, 10][[1, -1]]; len = Length[d]; EvenQ[len] && Union[Total[Partition[d, len/2]]] == {9}]]; Select[Range[300], ! okQ[#] &] (* T. D. Noe, Mar 02 2011 *)

Extensions

Corrected by T. D. Noe, Mar 02 2011