A175555 -id:A175555 - cp's OEIS frontend

A171869 a(n) is the period of A175555(n) in the sequence {A175555}.

1, 2, 2, 2, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296

Offset: 1

Michel Lagneau, Dec 30 2009, Apr 22 2010

			{A175555} = {5, 25, 375, 625, 59375, ...} =>
a(1) = 1 because A175555(1) = 5 occurs with the period 1 in the sequence A175555;
a(2) = 2 because A175555(2) = 25 occurs with the period 2 in the sequence A175555;
a(3) = 2 because A175555(3) = 375 occurs with the period 2 in the sequence A175555.

Cf. A002379, A170828, A175555, A170827.

nn:=2000:Digits:=nn:T:=array(1..nn):for n from 1 to nn do: T[n]:= irem(floor((1.5 ^n)*10^n),10^n): od: for a from 1 to nn do: z1:=T[a]: ii:=0:k:=0:for b from a+1 to nn while(ii)=0 do:k:=k+1:  z2:=irem(T[b],10^a): if z1=z2 then ii:=1:printf(`%d, `, k): else fi:od:od:

a(n) = 2^(n-4) if n > = 5.

More terms and edits. - Michel Lagneau, Jul 14 2012

A114205 Write decimal expansion of 1/n as 0.PPP...PQQQ..., where QQQ... is the cyclic part. If the expansion does not terminate, any leading 0's in QQQ... are regarded as being at the end of the PPP...P part. Sequence gives PPP...P, right justified, with leading zeros omitted.

Original entry on oeis.org

5, 0, 25, 2, 1, 0, 125, 0, 1, 0, 8, 0, 0, 0, 625, 0, 0, 0, 5, 0, 0, 0, 41, 4, 0, 0, 3, 0, 0, 0, 3125, 0, 0, 0, 2, 0, 0, 0, 25, 0, 0, 0, 2, 0, 0, 0, 208, 0, 2, 0, 1, 0, 0, 0, 17, 0, 0, 0, 1, 0, 0, 0, 15625, 0, 0, 0, 1, 0, 0, 0, 13, 0, 0, 1, 1, 0, 0, 0, 125, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0, 0, 10

Offset: 2

N. J. A. Sloane, Oct 17 2006

b(n) = A386406(n) gives the length of P (including leading zeros), c(n) = A036275(n) gives the smallest cycle in QQQ... (including terminating zeros) and d(n) = A051626(n) gives the length of that cycle.

Thus 1/n = 10^(-b(n)) * ( a(n) + c(n)/(10^d(n) - 1) ). When c(n)=d(n)=0, the fraction c(n)/(10^d(n) - 1), which is 0/0, evaluates (by definition) to 0.

			n .. expansion of 1/n .... a b c d
2 .50000000000000000000... 5 1 0 0
3 .33333333333333333333... 0 0 3 1
4 .25000000000000000000... 25 2 0 0
5 .20000000000000000000... 2 1 0 0
6 .16666666666666666667... 1 1 6 1
7 .14285714285714285714... 0 0 142857 6
8 .12500000000000000000... 125 3 0 0
9 .11111111111111111111... 0 0 1 1
10 .1000000000000000000... 1 1 0 0
11 .0909090909090909090... 0 1 90 2
12 .0833333333333333333... 8 2 3 1
13 .0769230769230769230... 0 1 769230 6
14 .0714285714285714285... 0 1 714285 6
15 .0666666666666666666... 0 1 6 1
16 .0625000000000000000... 625 4 0 0
(Start)
92 .0108695652173913043... 10  3 869...260 22
102 .009803921568627450... 0   2 980...450 16
416 .002403846153846153... 240 5 384615    6
4544 .00022007042253521... 2200 7 704...450 35
(End) - _Ruud H.G. van Tol_, Nov 20 2024

Cf. A386406, A114206, A036275, A051626, A060284, A007732, A175555.

A114205 := proc(n) local sh,lpow,mpow,a,b ; lpow:=1 ; while true do for mpow from lpow-1 to 0 by -1 do if (10^lpow-10^mpow) mod n =0 then a := (10^lpow-10^mpow)/n ; sh := 10^(lpow-mpow)-1 ; b := a mod sh ; a := floor(a/sh) ; while b>0 and b*10 < sh+1 do a := 10*a ; b := 10*b ; end ; RETURN(a) ; fi ; od ; lpow := lpow+1 ; od ; end: for n from 2 to 600 do printf("%d %d ",n,A114205(n)) ; od ; # R. J. Mathar, Oct 19 2006

fa[n_] := Block[{p},p = First[RealDigits[1/n]];If[ ! IntegerQ[Last[p]], p =  Join[Most[p],TakeWhile[Last[p],#==0&]]];FromDigits[p]];Table[fa[n], {n, 100}] (* Ray Chandler, Oct 18 2006 *)

a(n)= my(s=max(valuation(n, 2), valuation(n, 5))); s||return(0); my([p, r]= divrem(10^s, n)); if(r&&(r=n\r)>9, s+=logint(r, 10)); 10^s\n; \\ Ruud H.G. van Tol, Nov 19 2024

More terms from Ray Chandler and Hans Havermann, Oct 18 2006
I would also like to get programs that produce this and A114206, A036275, A051626 in Maple.
Edited by Andrei Zabolotskii, Jul 20 2025

A175557 Prime preperiodic part of the decimal expansion of 1/k as k runs through A065502.

Original entry on oeis.org

5, 2, 5, 41, 3, 2, 2, 2, 17, 13, 11, 89, 7, 5, 5, 5, 41, 3, 3, 347, 3, 3, 3, 29, 2, 2, 2, 2, 26041, 2, 2, 2, 23, 2, 2, 2, 2, 2, 17, 13, 13, 1201, 11, 11, 107, 919, 89, 7, 7, 7, 7, 7, 7, 61, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 41, 4111, 3

Offset: 1

Michel Lagneau, Jun 30 2010

Primes in A175555 in the order of appearance.
Multiples of 2 or 5 generate a quotient with a preperiodic sequence of digits, for example 1/24 = 0.041666666..., and 41 is the decimal form of the preperiodic part.
The corresponding values of n are: 2, 5, 20, 24, 28, 36, 44, 50, 56, 72, 88, 112, 136, 168, 184, ...

			The prime 347 is in the sequence because 1/288 = .00347222222222222222...
The prime 1201 is in the sequence because 1/832 =.001201 923076 923076 ...

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrueche'.

Cf. A175555, A036275, A065502.

for n from 1 do
    p := A175555(n) ;
    if isprime(p) then
        print(p) ;
    end if;
end do: # R. J. Mathar, Jul 22 2012

A179192 Numbers n, not relatively prime to 10, such that the decimal form of the period of 1/n is prime.

Original entry on oeis.org

12, 18, 30, 36, 45, 48, 75, 120, 180, 192, 198, 270, 288, 300, 330, 360, 450, 480, 495, 750, 768, 1152, 1200, 1584, 1800, 1875, 1920, 1980, 1998, 2304, 2700, 2880, 3000, 3072, 3300, 3330, 3600, 3690, 4500, 4800, 4950, 4995, 5625, 7500, 7680, 9090, 11520, 12000, 12288, 15840, 18000, 18432, 18750, 19200, 19800, 19980, 19998

Offset: 1

Michel Lagneau, Jul 01 2010

The sequence A175545 (numbers n such that the decimal form of the period of 1/n is prime) is only concerned with numbers n such that gcd(n,10)=1. Each number n such that gcd(n,10)<>1 generates a quotient where there exist a sequence of digits which is periodic after a finite sequence of digits, for example 1/36 = .0277777.... and 7 is periodic.
The prime numbers corresponding to this sequence are :
3, 5, 3, 7, 2, 3, 3, 3, 5, 3, 5, 37, 2, 3, 3, 7, 2, 3, 2,...

			1584 is in the sequence because 1/1584 = .0006313131313131313131... and 31 is prime.

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrueche'.

Cf. A175557, A175555, A178505, A045572, A002329

Reap[Do[p=RealDigits[1/n][[1,-1]]; If[GCD[10,n]>1 && Head[p] === List, While[p[[-1]] == 0, p=Most[p]]; If[PrimeQ[FromDigits[p]], Sow[n]]], {n, 20000}]][[2,1]]

Union of A179192 and A175545 is A061564.

Sequence corrected by T. D. Noe, Nov 18 2010

cp's OEIS Frontend

A171869 a(n) is the period of A175555(n) in the sequence {A175555}.

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A114205 Write decimal expansion of 1/n as 0.PPP...PQQQ..., where QQQ... is the cyclic part. If the expansion does not terminate, any leading 0's in QQQ... are regarded as being at the end of the PPP...P part. Sequence gives PPP...P, right justified, with leading zeros omitted.

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Maple

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A175557 Prime preperiodic part of the decimal expansion of 1/k as k runs through A065502.

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Keywords

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Crossrefs

Programs

Maple

A179192 Numbers n, not relatively prime to 10, such that the decimal form of the period of 1/n is prime.

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Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula

Extensions