A187040 Numbers for which Midy's theorem holds.
7, 11, 13, 14, 17, 19, 22, 23, 26, 28, 29, 34, 35, 38, 44, 46, 47, 49, 52, 55, 56, 58, 59, 61, 65, 68, 70, 73, 76, 77, 85, 88, 89, 91, 92, 94, 95, 97, 98, 101, 103, 104, 109, 110, 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
Offset: 1
Links
- Gilberto García-Pulgarín and Hernán Giraldo, Characterizations of Midy's property, Integers 9 (2009), A18, 191--197. MR2506150 (2010f:11013).
- Joseph Lewittes, Midy's theorem for periodic decimals, arXiv:math/0605182 [math.NT], 2006.
- Harold W. Martin, Generalizations of midy’s theorem on repeating decimals, INTEGERS 7 (2007), #A03.
- Étienne Midy, De quelques propriétés des nombres et des fractions décimales périodiques, 1836.
- Wikipedia, Midy's theorem
Programs
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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: A187040 := proc(n) local st: st := op(4,numtheory[pdexpand](1/n)); if (modp(nops(st),2) = 0) then if (10^(nops(st)/2)-1 - (fct1(st)+fct2(st)) = 0) then RETURN(n) fi: fi: end: seq(A187040(n), n=2..200);
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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[200], okQ] (* T. D. Noe, Mar 02 2011 *)
Extensions
Corrected by T. D. Noe, Mar 02 2011
Comments