A007732 -id:A007732 - cp's OEIS frontend

A002371 Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).

0, 1, 0, 6, 2, 6, 16, 18, 22, 28, 15, 3, 5, 21, 46, 13, 58, 60, 33, 35, 8, 13, 41, 44, 96, 4, 34, 53, 108, 112, 42, 130, 8, 46, 148, 75, 78, 81, 166, 43, 178, 180, 95, 192, 98, 99, 30, 222, 113, 228, 232, 7, 30, 50, 256, 262, 268, 5, 69, 28, 141, 146, 153, 155, 312, 79, 110

Offset: 1

N. J. A. Sloane

a(n) is the minimum solution x of modular equation 10^x == 1 (mod p), where p = prime(n). - Carmine Suriano, Oct 10 2012
a(n) = smallest m such that 111...11 (m 1's) is divisible by the n-th prime, or 0 if no such m exists (with the exception that a(2) = 3 instead of 1). E.g., the 5th prime, 11, divides 11, so a(5) = 2. - N. J. A. Sloane, Oct 03 2013 [Comment corrected by Derek Orr, Jun 14 2014]
Numbers n such that A071126(n) = A000040(n) - 1. - Hugo Pfoertner, Mar 18 2003
Except for n = 1 and 3, a(n) divides A006093(n). - Robert Israel, Jul 15 2016

			A002371(11) = 15 because the 11th prime is 31, and 1/31 = 0.03225806451612903225806451612903225806452... has period 15. - _Richard F. Lyon_, Mar 29 2022

Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed. New York: Dover, 1966, pages 65, 309. ISBN 0-486-21096-0.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, 1996, p. 162. ISBN 978-0-387-97993-9.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
C. K. Caldwell, The Prime Glossary, Period of a prime
Matt Parker and Brady Haran, The Reciprocals of Primes, Numberphile video (2022).
William Shanks, On the number of figures in the period of the reciprocal of every prime number below 20,000, Proc. Royal Soc. London, 22 (1874), 200-210. See also on JSTOR.
Eric Weisstein's World of Mathematics, Decimal Expansion
Index entries for sequences related to decimal expansion of 1/n

See A048595 for another version. Cf. A006883, A007732, A051626, A071126, A000040, A002275, A097443.
Cf. A001913 (full repetend primes), A060257 (1/prime(n) has period prime(n) - 1).
In other bases: A014664, A062117, A211241, A211242, A211243, A211244, A211245, A002371.

seq(subs(FAIL=0,numtheory:-order(10, ithprime(n))),n=1..100); # Robert Israel, Jul 15 2016

Table[ Length[ RealDigits[1 / Prime[n]] [[1, 1]]], {n, 1, 70}]
Table[If[IntegerQ[#], #, 0] &[MultiplicativeOrder[10, Prime[n]]], {n, 1, 70}] (* Jan Mangaldan, Jul 07 2020 *)

a(n)=if(n<4,n==2,znorder(Mod(10, prime(n))))

from sympy import prime, n_order
def A002371(n): return 0 if n == 1 or n == 3 else n_order(10,prime(n)) # Chai Wah Wu, Feb 07 2022

From Alexander Adamchuk, Jan 28 2007: (Start)
a(A000720(p)) = p - 1 for primes p in A001913.
a(A060257(n)) = prime(A060257(n)) - 1. (End)

More terms from Arlin Anderson (starship1(AT)gmail.com)

Edited by Charles R Greathouse IV, Mar 24 2010

A051626 Period of decimal representation of 1/n, or 0 if 1/n terminates.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, 6, 6, 1, 0, 16, 1, 18, 0, 6, 2, 22, 1, 0, 6, 3, 6, 28, 1, 15, 0, 2, 16, 6, 1, 3, 18, 6, 0, 5, 6, 21, 2, 1, 22, 46, 1, 42, 0, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 0, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13, 0, 9, 5, 41, 6, 16, 21, 28, 2, 44, 1

Offset: 1

J. Lowell

Essentially same as A007732.

For any prime number p: if a(p) > 0, a(p) divides p-1. - David Spitzer, Jan 09 2017

			From _M. F. Hasler_, Dec 14 2015: (Start)
a(1) = a(2) = 0 because 1/1 = 1 and 1/2 = 0.5 have a finite decimal expansion.
a(3) = a(6) = a(9) = a(12) = 1 because 1/3 = 0.{3}*, 1/6 = 0.1{6}*, 1/9 = 0.{1}*, 1/12 = 0.08{3}* where the sequence of digits {...}* which repeats indefinitely is of length 1.
a(7) = 6 because 1/7 = 0.{142857}* with a period of 6.
a(17) = 16 because 1/17 = 0.{0588235294117647}* with a period of 16.
a(19) = 18 because 1/19 = 0.{052631578947368421}* with a period of 18. (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
Project Euler, Reciprocal cycles: Problem 26
Eric Weisstein's World of Mathematics, Repeating Decimal
Index entries for sequences related to decimal expansion of 1/n

Essentially same as A007732. Cf. A002371, A048595, A006883, A036275, A114205, A114206, A001913.

A051626 := proc(n) local lpow,mpow ;
    if isA003592(n) then
       RETURN(0) ;
    else
       lpow:=1 ;
       while true do
          for mpow from lpow-1 to 0 by -1 do
              if (10^lpow-10^mpow) mod n =0 then
                 RETURN(lpow-mpow) ;
              fi ;
          od ;
          lpow := lpow+1 ;
       od ;
    fi ;
end: # R. J. Mathar, Oct 19 2006

r[x_]:=RealDigits[1/x]; w[x_]:=First[r[x]]; f[x_]:=First[w[x]]; l[x_]:=Last[w[x]]; z[x_]:=Last[r[x]];
d[x_] := Which[IntegerQ[l[x]], 0, IntegerQ[f[x]]==False, Length[f[x]], True, Length[l[x]]]; Table[d[i], {i,1,90}] (* Hans Havermann, Oct 19 2006 *)
fd[n_] := Block[{q},q = Last[First[RealDigits[1/n]]];If[IntegerQ[q], q = {}]; Length[q]];Table[fd[n], {n, 100}] (* Ray Chandler, Dec 06 2006 *)
Table[Length[RealDigits[1/n][[1,-1]]],{n,90}] (* Harvey P. Dale, Jul 03 2011 *)
a[n_] := If[ PowerMod[10, n, n] == 0, 0, MultiplicativeOrder[10, n/2^IntegerExponent[n, 2]/5^IntegerExponent[n, 5]]]; Array[a, 90] (* myself in A003592 and T. D. Noe in A007732 *) (* Robert G. Wilson v, Feb 20 2025 *)

A051626(n)=if(1M. F. Hasler, Dec 14 2015

def A051626(n):
    if isA003592(n):
        return 0
    else:
        lpow=1
        while True:
            for mpow in range(lpow-1,-1,-1):
                if (10**lpow-10**mpow) % n == 0:
                    return lpow-mpow
            lpow += 1 # Kenneth Myers, May 06 2016

from sympy import multiplicity, n_order
def A051626(n): return 0 if (m:=(n>>(~n & n-1).bit_length())//5**multiplicity(5,n)) == 1 else n_order(10,m) # Chai Wah Wu, Aug 11 2022

a(n)=A132726(n,1); a(n)=a(A132740(n)); a(A132741(n))=a(A003592(n))=0. - Reinhard Zumkeller, Aug 27 2007

More terms from James Sellers

A036275 The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.

Original entry on oeis.org

0, 0, 3, 0, 0, 6, 142857, 0, 1, 0, 90, 3, 769230, 714285, 6, 0, 5882352941176470, 5, 526315789473684210, 0, 476190, 45, 4347826086956521739130, 6, 0, 384615, 370, 571428, 3448275862068965517241379310, 3, 322580645161290, 0, 30, 2941176470588235, 285714, 7

Offset: 1

Floor van Lamoen

a(n) = 0 iff n = 2^i*5^j (A003592). - Jon Perry, Nov 19 2014

a(n) = n iff n = 3 or 6 (see De Koninck & Mercier reference). - Bernard Schott, Dec 02 2020

			1/28 = .03571428571428571428571428571428571428571... and digit-cycle is 571428, so a(28)=571428.

Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 347 pp. 50 and 205, Ellipses, Paris, 2004.

Philippe Guglielmetti, Table of n, a(n) for n = 1..1001 (first 500 terms from T. D. Noe)
Index entries for sequences related to decimal expansion of 1/n

Cf. A007732, A051626, A002371, A048595, A006883, A007498, A007615, A040017, A051627.
See also A060282, A060283, A060251.
A051628 is length of preamble.

isCycl := proc(n) local ifa,i ; if n <= 2 then RETURN(false) ; fi ; ifa := ifactors(n)[2] ; for i from 1 to nops(ifa) do if op(1,op(i,ifa)) <> 2 and op(1,op(i,ifa)) <> 5 then RETURN(true) ; fi ; od ; RETURN(false) ; end: A036275 := proc(n) local ifa,sh,lpow,mpow,r ; if not isCycl(n) then RETURN(0) ; else lpow:=1 ; while true do for mpow from lpow-1 to 0 by -1 do if (10^lpow-10^mpow) mod n =0 then r := (10^lpow-10^mpow)/n ; r := r mod (10^(lpow-mpow)-1) ; while r*10 < 10^(lpow-mpow) do r := 10*r ; od ; RETURN(r) ; fi ; od ; lpow := lpow+1 ; od ; fi ; end: for n from 1 to 60 do printf("%d %d ",n,A036275(n)) ; od ; # R. J. Mathar, Oct 19 2006

fc[n_]:=Block[{q=RealDigits[1/n][[1,-1]]},If[IntegerQ[q],0,While[First[q]==0,q=RotateLeft[q]];FromDigits[q]]];
Table[fc[n],{n,36}] (* Ray Chandler, Nov 19 2014, corrected Jun 27 2017 *)
Table[FromDigits[FindTransientRepeat[RealDigits[1/n,10,120][[1]],3] [[2]]],{n,40}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 12 2019 *)

Corrected and extended by N. J. A. Sloane

Corrected a(92), a(208), a(248), a(328), a(352) and a(488) which missed a trailing zero (see the table). - Philippe Guglielmetti, Jun 20 2017

A097443 Half-period primes, i.e., primes p for which the decimal expansion of 1/p has period (p-1)/2.

Original entry on oeis.org

3, 13, 31, 43, 67, 71, 83, 89, 107, 151, 157, 163, 191, 197, 199, 227, 283, 293, 307, 311, 347, 359, 373, 401, 409, 431, 439, 443, 467, 479, 523, 557, 563, 569, 587, 599, 601, 631, 653, 677, 683, 719, 761, 787, 827, 839, 877, 881, 883, 911, 919, 929, 947, 991

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Aug 23 2004

nonn

Primes p for which 10 has multiplicative order (p-1)/2. - Robert Israel, Jul 15 2016

			13 is a half-period prime because 1/13 = 0.076923076923076923076923..., which has period 6, or (13-1)/2.

Robert Israel, Table of n, a(n) for n = 1..10000 (Terms 2..1001 from T. D. Noe.)
Makoto Kamada, Factorizations of 11...11 (Repunit).
Index entries for sequences related to decimal expansion of 1/n

Almost the same as A001914.
Cf. A001913, A055628, A056157, A056210-A056217, A098680
Cf. also A000040, A007732, A006883, A097443, A097955.

select(t -> isprime(t) and numtheory:-order(10, t) = (t-1)/2,
[seq(t,t = 3..1000,2)]); # Robert Israel, Jul 15 2016

f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 200]], f[ # ] == 2 &] (* Robert G. Wilson v, Sep 14 2004 *)

is(n)= gcd(10,n)==1 && isprime(n) && znorder(Mod(10,n))==(n-1)/2 \\ Dana Jacobsen, Jul 19 2016

use ntheory ":all"; forprimes { say if znorder(10,$) == ($-1)/2; } 1,1000; # Dana Jacobsen, Jul 19 2016

Edited (including prepending 3), at the suggestion of Georg Fischer, by N. J. A. Sloane, Oct 19 2018

A040017 Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime) of the form A019328(r)/gcd(A019328(r),r) in order (periods r are given in A051627).

Original entry on oeis.org

3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991, 909090909090909090909090909091

Offset: 1

Jud McCranie

Prime p=3 is the only known example of a unique period prime such that A019328(r)/gcd(A019328(r),r) = p^k with k > 1 (cf. A323748). It is plausible to assume that no other such prime exists. Under this (unproved) assumption, the current sequence lists all unique period primes in order and represents a sorted version of A007615. - Max Alekseyev, Oct 14 2022

			The decimal expansion of 1/101 is 0.00990099..., having a period of 4 and it is the only prime with that period.

J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 324, Pour la Science Paris 2000.

Robert G. Wilson v, Table of n, a(n) for n = 1..47
Chris Caldwell, The Prime Glossary, Unique prime.
C. K. Caldwell, "Top Twenty" page, Unique.
Chris K. Caldwell and Harvey Dubner, Unique-Period Primes, J. Recreational Math., 29:1 (1998) 43-48.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Unique Prime.
Wikipedia, Unique prime.
Index entries for sequences related to decimal expansion of 1/n

Cf. A007615, A007498, A002371, A048595, A006883, A007732, A051626, A051627, A323748.

lst = {}; Do[c = Cyclotomic[n, 10]; q = c/GCD[c, n]; If[PrimeQ[q], AppendTo[lst, q]], {n, 62}]; Prepend[Sort[lst], 3] (* Arkadiusz Wesolowski, May 13 2012 *)

For n >= 2, a(n) = A019328(r) / gcd(A019328(r), r), where r = A051627(n). - Max Alekseyev, Oct 14 2022

Missing term a(45) inserted in b-file at the suggestion of Eric Chen by Max Alekseyev, Oct 13 2022

Edited by Max Alekseyev, Oct 14 2022

A054710 Number of powers of 10 mod n.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 6, 4, 1, 2, 2, 3, 6, 7, 2, 5, 16, 2, 18, 3, 6, 3, 22, 4, 3, 7, 3, 8, 28, 2, 15, 6, 2, 17, 7, 3, 3, 19, 6, 4, 5, 7, 21, 4, 2, 23, 46, 5, 42, 3, 16, 8, 13, 4, 3, 9, 18, 29, 58, 3, 60, 16, 6, 7, 7, 3, 33, 18, 22, 7, 35, 4, 8, 4, 3, 20, 6, 7, 13, 5, 9, 6, 41, 8, 17, 22, 28, 5, 44, 2

Offset: 1

Henry Bottomley, Apr 20 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A007732, A051628, A121341.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

Table[Length[Union[PowerMod[10, Range[0,n], n]]], {n,100}] (* T. D. Noe, Aug 30 2006 *)
a[n_] := Module[{e = IntegerExponent[n, {2, 5}]}, Max[e] + MultiplicativeOrder[10, n/Times @@ ({2, 5}^e)]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007732(n) + A051628(n). - Amiram Eldar, Aug 25 2024

A132740 Largest divisor of n coprime to 10.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 11, 3, 13, 7, 3, 1, 17, 9, 19, 1, 21, 11, 23, 3, 1, 13, 27, 7, 29, 3, 31, 1, 33, 17, 7, 9, 37, 19, 39, 1, 41, 21, 43, 11, 9, 23, 47, 3, 49, 1, 51, 13, 53, 27, 11, 7, 57, 29, 59, 3, 61, 31, 63, 1, 13, 33, 67, 17, 69, 7, 71, 9, 73, 37, 3, 19, 77, 39, 79, 1, 81

Offset: 1

Reinhard Zumkeller, Aug 27 2007

Or: n with all factors of 2 and 5 removed. - M. F. Hasler, Apr 25 2017

			a(1050) = a(2*3*5*5*7) = 3*7 = 21.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000265, A003592, A069105, A070021, A070022, A070023, A132739, A132741.

a132740 = a132739 . a000265  -- Reinhard Zumkeller, Apr 08 2011

A132740 := proc(n) n/A132741(n) ; end proc: # R. J. Mathar, Sep 06 2011

a[n_] := FixedPoint[ Quotient[#, GCD[#, 10]]& , n]; Table[a[n], {n, 1, 81}] (* Jean-François Alcover, Sep 06 2011, after Vladimir Joseph Stephan Orlovsky *)
Table[SelectFirst[Reverse[Divisors[n]],CoprimeQ[#,10]&],{n,90}] (* Uses the SelectFirst function from Mathematica version 10. - Harvey P. Dale, Mar 22 2015 *)
a[n_] := n / Times @@ ({2, 5}^IntegerExponent[n, {2, 5}]); Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

a(n)=n/5^valuation(n,5)>>valuation(n,2) \\ Charles R Greathouse IV, Sep 06 2011

a(n) = A000265(A132739(n)) = A132739(A000265(n)) = n / A132741(n);
A051626(a(n)) = A051626(n); A007732(a(n)) = A007732(n);
a(A003592(n)) = 1.
Multiplicative with a(2^e) = 1, a(5^e) = 1 and a(p^e) = p^e for p = 3 and p >= 7.
Dirichlet g.f. zeta(s-1)*(2^s-2)*(5^s-5)/((2^s-1)*(5^s-1)). - R. J. Mathar, Sep 06 2011
Sum_{k=1..n} a(k) ~ (5/18) * n^2. - Amiram Eldar, Nov 28 2022

Edited by M. F. Hasler, Apr 25 2017

A007498 Unique period lengths of primes mentioned in A007615.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 294, 317, 320, 385, 586, 597, 654, 738, 945, 1031, 1172, 1282, 1404, 1426, 1452, 1521, 1752, 1812, 1836, 1844, 1862, 2134, 2232, 2264, 2667, 3750, 3903, 3927, 4274, 4354

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Let {Zs(m, 10, 1)} be the Zsigmondy numbers for a = 10, b = 1: Zs(m, 10, 1) is the greatest divisor of 10^m - 1^m that is coprime to 10^r - 1^r for all positive integers r < m. Then this sequence gives m such that Zs(m, 10, 1) is a prime power (e.g., Zs(1, 10, 1) = 9 = 3^2, Zs(2, 10, 1) = 11, Zs(3, 10, 1) = 37, Zs(4, 10, 1) = 101). It is very likely that Zs(m, 10, 1) is prime if m > 1 is in this sequence (note that the Mathematica and PARI programs below are based on this assumption). - Jianing Song, Aug 12 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 22-24.

Ray Chandler, Table of n, a(n) for n = 1..106
Chris K. Caldwell, Unique (period) primes and the factorization of cyclotomic polynomials minus one, Mathematica Japonica, 26 (1997), 189-195.
Chris K. Caldwell & H. Dubner, Unique-Period Primes, Table 2 in Journal of Recreational Mathematics 29(1) 46 1998.
Robert G. Wilson v, Notes, n.d.
Index entries for sequences related to decimal expansion of 1/n

Cf. A007615, A002371, A048595, A006883, A007732, A051626, A046108.

Cf. A161508 (unique period lengths in base 2).

lst={1}; Do[p=Cyclotomic[n, 10]/GCD[n, Cyclotomic[n, 10]]; If[PrimeQ[p], AppendTo[lst, n]], {n, 3000}]; lst (* T. D. Noe, Sep 08 2005 *)

isok(n) = if (n==1, 1, my(p = polcyclo(n, 10)); isprime(p/gcd(p, n))); \\ Michel Marcus, Jun 20 2018

More terms from T. D. Noe, Sep 08 2005

a(48)-a(52) from Ray Chandler, Jul 09 2008

A084680 Order of 10 modulo n [i.e., least m such that 10^m = 1 (mod n)] or 0 when no such number exists.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 6, 0, 1, 0, 2, 0, 6, 0, 0, 0, 16, 0, 18, 0, 6, 0, 22, 0, 0, 0, 3, 0, 28, 0, 15, 0, 2, 0, 0, 0, 3, 0, 6, 0, 5, 0, 21, 0, 0, 0, 46, 0, 42, 0, 16, 0, 13, 0, 0, 0, 18, 0, 58, 0, 60, 0, 6, 0, 0, 0, 33, 0, 22, 0, 35, 0, 8, 0, 0, 0, 6, 0, 13, 0, 9, 0, 41, 0, 0, 0, 28, 0, 44, 0, 6, 0, 15, 0, 0, 0

Offset: 1

Lekraj Beedassy, Jun 30 2003

nonn

When n is not divisible by 2 or 5, a(n) = A007732(n). A002329 contains the nonzero terms.

A number k > 0 appears in this sequence exactly A059892(k) times. - T. D. Noe, May 18 2007

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A007732, A002329, A070682, A216415.

List([1..100],n->OrderMod(10,n)); # Muniru A Asiru, Feb 26 2019

A084680 := proc(n) if gcd(n,10) <> 1 then 0 ; elif n = 1 then 1 ; else numtheory[order](10,n) ; end if; end proc: seq(A084680(n),n=2..100) ; # R. J. Mathar, Mar 10 2010

a[n_] := If[!CoprimeQ[n, 10], 0, MultiplicativeOrder[10, n]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 07 2012 *)

a(n,b=10)=if(gcd(n,b)!=1,0,znorder(Mod(b,n)));
vector(66,n,a(n)) \\ Joerg Arndt, Nov 15 2014

More terms from John W. Layman, Aug 12 2003

A132741 Largest divisor of n having the form 2^i*5^j.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 1, 2, 5, 16, 1, 2, 1, 20, 1, 2, 1, 8, 25, 2, 1, 4, 1, 10, 1, 32, 1, 2, 5, 4, 1, 2, 1, 40, 1, 2, 1, 4, 5, 2, 1, 16, 1, 50, 1, 4, 1, 2, 5, 8, 1, 2, 1, 20, 1, 2, 1, 64, 5, 2, 1, 4, 1, 10, 1, 8, 1, 2, 25, 4, 1, 2, 1, 80, 1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 1, 2, 5, 32, 1, 2

Offset: 1

Reinhard Zumkeller, Aug 27 2007

The range of this sequence, { a(n); n>=0 }, is equal to A003592. - M. F. Hasler, Dec 28 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003592, A006519, A007732, A007814, A051626, A060904, A112765, A132740.
Cf. A001620, A082633.
Cf. A379003 (ordinal transform), A379004 (rgs-transform).
Cf. also A355582.

a132741 = f 2 1 where
   f p y x | r == 0    = f p (y * p) x'
           | otherwise = if p == 2 then f 5 y x else y
           where (x', r) = divMod x p
-- Reinhard Zumkeller, Nov 19 2015

A132741 := proc(n) local f,a; f := ifactors(n)[2] ; a := 1; for f in ifactors(n)[2] do if op(1,f) =2 then a := a*2^op(2,f) ; elif op(1,f) =5 then a := a*5^op(2,f) ; end if; end do;a; end proc: # R. J. Mathar, Sep 06 2011

a[n_] := SelectFirst[Reverse[Divisors[n]], MatchQ[FactorInteger[#], {{1, 1}} | {{2, }} | {{5, }} | {{2, }, {5, }}]&]; Array[a, 100] (* Jean-François Alcover, Feb 02 2018 *)
a[n_] := Times @@ ({2, 5}^IntegerExponent[n, {2, 5}]); Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

A132741(n)=5^valuation(n,5)<M. F. Hasler, Dec 28 2015

a(n) = n / A132740(n).
a(A003592(n)) = A003592(n).
A051626(a(n)) = 0.
A007732(a(n)) = 1.
From R. J. Mathar, Sep 06 2011: (Start)
Multiplicative with a(2^e)=2^e, a(5^e)=5^e and a(p^e)=1 for p=3 or p>=7.
Dirichlet g.f. zeta(s)*(2^s-1)*(5^s-1)/((2^s-2)*(5^s-5)). (End)
a(n) = A006519(n)*A060904(n) = 2^A007814(n)*5^A112765(n). - M. F. Hasler, Dec 28 2015
Sum_{k=1..n} a(k) ~ n*(12*log(n)^2 + (24*gamma + 36*log(2) - 24)*log(n) + 24 - 24*gamma - 36*log(2) + 36*gamma*log(2) + 2*log(2)^2 - 18*log(5) + 18*gamma*log(5) + 27*log(2)*log(5) + 2*log(5)^2 + 18*log(5)*log(n) - 24*gamma_1)/(60*log(2)*log(5)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Jan 26 2023

cp's OEIS Frontend

A002371 Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).

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A051626 Period of decimal representation of 1/n, or 0 if 1/n terminates.

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A036275 The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.

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A097443 Half-period primes, i.e., primes p for which the decimal expansion of 1/p has period (p-1)/2.

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A040017 Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime) of the form A019328(r)/gcd(A019328(r),r) in order (periods r are given in A051627).

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A054710 Number of powers of 10 mod n.

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