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Numbers for which Midy's theorem holds.
3

%I #35 Jan 09 2024 02:59:40

%S 7,11,13,14,17,19,22,23,26,28,29,34,35,38,44,46,47,49,52,55,56,58,59,

%T 61,65,68,70,73,76,77,85,88,89,91,92,94,95,97,98,101,103,104,109,110,

%U 112,113,115,116,118,121,122,127,130,131,133,136,137,139,140,143,145,146,149,152,154,157,161,167,169,170,175,176,178,179,181,182,184,188,190,193,194,196,197

%N Numbers for which Midy's theorem holds.

%C Martin gives generalizations of Midy's theorem that characterize the numbers in this sequence. See theorem 8. - _T. D. Noe_, Mar 02 2011

%C García-Pulgarín Gilberto and Giraldo Hernán give the characterization of the numbers that satisfy Midy's property.

%H Gilberto García-Pulgarín and Hernán Giraldo, <a href="http://www.emis.de/journals/INTEGERS/papers/j18/j18.Abstract.html">Characterizations of Midy's property,</a> Integers 9 (2009), A18, 191--197. MR2506150 (2010f:11013).

%H Joseph Lewittes, <a href="http://arxiv.org/abs/math/0605182">Midy's theorem for periodic decimals</a>, arXiv:math/0605182 [math.NT], 2006.

%H Harold W. Martin, <a href="http://www.emis.de/journals/INTEGERS/papers/h3/h3.Abstract.html">Generalizations of midy’s theorem on repeating decimals</a>, INTEGERS 7 (2007), #A03.

%H Étienne Midy, <a href="https://archive.org/details/1014596483">De quelques propriétés des nombres et des fractions décimales périodiques</a>, 1836.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Midy%27s_theorem">Midy's theorem</a>

%p fct1 := proc(an) local i,st: st := 0:

%p for i from 1 to nops(an)/2 do

%p st := op(i,an)*10^(nops(an)/2-i) + st

%p od: RETURN(st): end:

%p fct2 := proc(an) local i,st: st := 0:

%p for i from nops(an)/2+1 to nops(an) do

%p st := op(i,an)*10^(nops(an)/2-i+nops(an)/2) + st

%p od: RETURN(st): end:

%p A187040 := proc(n) local st:

%p st := op(4,numtheory[pdexpand](1/n));

%p if (modp(nops(st),2) = 0) then

%p if (10^(nops(st)/2)-1 - (fct1(st)+fct2(st)) = 0) then

%p RETURN(n)

%p fi: fi: end: seq(A187040(n), n=2..200);

%t 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[200], okQ] (* _T. D. Noe_, Mar 02 2011 *)

%Y Cf. A028416, A187041.

%K nonn,base

%O 1,1

%A _Jani Melik_, Mar 02 2011

%E Corrected by _T. D. Noe_, Mar 02 2011