A187040 -id:A187040 - cp's OEIS frontend

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

Jani Melik, Mar 02 2011

Wikipedia, Midy's theorem

Cf. A028416, A187040.

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);

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 *)

Corrected by T. D. Noe, Mar 02 2011

A216664 Odd numbers n such that the decimal expansion of 1/n contains the digit "9" at position (n + 1)/2.

Original entry on oeis.org

11, 17, 19, 23, 29, 47, 59, 61, 73, 91, 95, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 167, 179, 181, 189, 193, 211, 223, 229, 233, 251, 255, 257, 263, 269, 313, 325, 331, 337, 349, 353, 367, 379, 383, 389, 419, 421, 433, 441, 457, 461, 463, 477, 487, 491

Offset: 1

Arkadiusz Wesolowski, Sep 14 2012

First nine terms are primes.

This is not a subsequence of A187040: 189 belongs to this sequence but not to A187040.

			1/17 = .058823529..., therefore 17 is a term.
1/21 = .04761904761..., therefore 21 is not a term.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Wikipedia, Midy's theorem
Index entries for sequences related to decimal expansion of 1/n

Cf. A187040.

lst = {}; Do[l = (n + 1)/2; d = Flatten@RealDigits[1/n, 10, l]; If[Join[Table[0, {-1*Last@d}], Most@d][[l]] == 9, AppendTo[lst, n]], {n, 1, 491, 2}]; lst

forstep(n=1, 491, 2, s=(n+1)/2; "\p s"; if(Mod(floor(10^s/n), 10)==9, print1(n, ", "))); \\ Arkadiusz Wesolowski, Aug 23 2013

from itertools import count, islice
def A216664_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue+1-startvalue%2,1),2):
        if 10**((n+1)//2)//n % 10 == 9:
            yield n
A216664_list = list(islice(A216664_gen(),20)) # Chai Wah Wu, Mar 04 2022

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

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A216664 Odd numbers n such that the decimal expansion of 1/n contains the digit "9" at position (n + 1)/2.

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