Displaying 1-10 of 11 results found.
Primes with unique period length (the periods are given in A007498).
(Formerly M2890)
+20
8
3, 11, 37, 101, 333667, 9091, 9901, 909091, 1111111111111111111, 11111111111111111111111, 99990001, 999999000001, 909090909090909091, 900900900900990990990991, 9999999900000001, 909090909090909090909090909091, 900900900900900900900900900900990990990990990990990990990991
COMMENTS
Additional terms are Phi(n,10)/gcd(n,Phi(n,10)) for the n in A007498, where Phi(n,10) is the n-th cyclotomic polynomial evaluated at 10.
REFERENCES
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.
EXAMPLE
3 is the only prime p such that decimal expansion of 1/p has (nontrivial) period exactly 1.
MATHEMATICA
nmax = 50; periods = Reap[ Do[ p = Cyclotomic[n, 10] / GCD[n, Cyclotomic[n, 10]]; If[ PrimeQ[p], Sow[n]], {n, 1, nmax}]][[2, 1]]; Cyclotomic[#, 10] / GCD[#, Cyclotomic[#, 10]]& /@ periods // Prepend[#, 3]& (* Jean-François Alcover, Mar 28 2013 *)
Indices n such that A019328(n) = Phi(n,10) is prime, where Phi is a cyclotomic polynomial.
+10
25
2, 4, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 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, 5877, 6022
COMMENTS
Unique period primes ( A040017) are often of the form Phi(k,10) or Phi(k,-10). Terms of this sequence which are the square of a prime, a(n)=p^2, are such that A252491(p) is prime. Apart from a(2)=2^2, there is no such term up to 26570. - M. F. Hasler, Jan 09 2015
MATHEMATICA
Select[Range[1000], PrimeQ[Cyclotomic[#, 10]] &] (* T. D. Noe, Mar 03 2012 *)
PROG
(PARI) for( i=1, 999, isprime( polcyclo(i, 10)) && print1( i", "))
EXTENSIONS
a(51)-a(91) from Ray Chandler, Maksym Voznyy et al. (cf. Phi_n(10) link), ca. 2009
The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.
+10
21
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
COMMENTS
a(n) = n iff n = 3 or 6 (see De Koninck & Mercier reference). - Bernard Schott, Dec 02 2020
REFERENCES
Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 347 pp. 50 and 205, Ellipses, Paris, 2004.
EXAMPLE
1/28 = .03571428571428571428571428571428571428571... and digit-cycle is 571428, so a(28)=571428.
MAPLE
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
MATHEMATICA
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 *)
EXTENSIONS
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
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).
+10
17
3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991, 909090909090909090909090909091
COMMENTS
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
REFERENCES
J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 324, Pour la Science Paris 2000.
LINKS
C. K. Caldwell, "Top Twenty" page, Unique. Chris K. Caldwell and Harvey Dubner, Unique-Period Primes, J. Recreational Math., 29:1 (1998) 43-48.
EXAMPLE
The decimal expansion of 1/101 is 0.00990099..., having a period of 4 and it is the only prime with that period.
MATHEMATICA
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 *)
Number of primitive prime factors of 10^n-1.
+10
9
1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 3, 3, 1, 1, 3, 2, 2, 3, 5, 3, 3, 5, 2, 3, 3, 1, 3, 1, 1, 2, 4, 3, 4, 3, 2, 4, 2, 1, 2, 3, 4, 2, 4, 2, 4, 2, 3, 2, 2, 3, 7, 1, 5, 4, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 4, 4, 6, 2, 5, 2, 3, 2, 3, 3, 3, 2, 5, 3, 7, 3, 1, 3, 5, 4, 3, 2, 4, 4
COMMENTS
Also the number of primes whose reciprocal is a repeating decimal of length n. The number of numbers in each row of table A046107. By Zsigmondy's theorem, a(n) >= 1. When a(n)=1, the corresponding prime is called a unique prime (see A007498, A040017 and A051627).
MATHEMATICA
pp={}; Table[f=Transpose[FactorInteger[10^n-1]][[1]]; p=Complement[f, pp]; pp=Union[pp, p]; Length[p], {n, 66}]
CROSSREFS
Cf. A007138 (smallest primitive prime factor of 10^n-1), A102347 (number of distinct prime factors of 10^n-1), A046107.
EXTENSIONS
Terms to a(276) in b-file from T. D. Noe, Jun 01 2010
Numbers k such that 2^k-1 has only one primitive prime factor.
+10
9
2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208
COMMENTS
Also, numbers k such that A086251(k) = 1. Also, numbers k such that A064078(k) is a prime power. The corresponding primitive primes are listed in A161509. The binary expansion of 1/p has period k and this is the only prime with such a period. The binary analog of A007498. This sequence has many terms in common with A072226. A072226 has the additional term 6; but it does not have terms 18, 20, 21, 54, 147, 342, 602, and 889 (less than 10000).
MATHEMATICA
Select[Range[1000], PrimePowerQ[Cyclotomic[ #, 2]/GCD[Cyclotomic[ #, 2], # ]]&]
1, 2, 3, 4, 10, 12, 9, 14, 24, 36, 48, 38, 19, 23, 39, 62, 120, 150, 106, 93, 134, 294, 196, 320, 654, 738, 385, 586, 317, 597, 1404, 945, 1452, 1836, 1752, 1172, 1812, 1282, 1426, 2232, 1862, 1844, 1521, 2134, 3750, 1031, 2264, 2667, 4354, 3927, 4274, 6522, 3903, 6022, 6682, 6135, 9550, 5877
COMMENTS
The numbers in A007498 sorted according to the magnitude of the corresponding prime. - T. D. Noe, Sep 08 2005
EXAMPLE
The decimal expansion of 1/101 is 0.00990099..., having a period of 4 and it is the only prime with that period.
MATHEMATICA
nmax = 10000; primesPeriods = Reap[Do[p = Cyclotomic[n, 10]/GCD[n, Cyclotomic[n, 10]]; If[PrimeQ[p], Print[n]; Sow[{p, n}]], {n, 1, nmax}]][[2, 1]]; Sort[primesPeriods][[All, 2]] // Prepend[#, 1]& // Take[#, 58]& (* Jean-François Alcover, Mar 29 2013 *)
EXTENSIONS
Corrected a(45)=3750 and extended by Ray Chandler, Oct 13 2008
Decimal period of 1/b(n), where b(n) is A046107.
+10
2
1, 2, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 28, 29, 29, 29, 29, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 32, 32, 33, 33, 34, 34, 34, 35, 35
Numbers k such that prime(k) does not contain the digit 1.
+10
2
1, 2, 3, 4, 9, 10, 12, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 48, 49, 50, 51, 52, 55, 56, 57, 59, 61, 62, 63, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 84, 85, 86, 87, 88, 90, 91, 92, 93, 95, 96, 97, 99, 101, 102, 103, 104, 106, 107, 108, 109, 111, 117, 118, 119, 120
EXAMPLE
99 is a term because prime(99) = 523 is unit-free.
MATHEMATICA
Select[ Range[120], Count[ IntegerDigits[ Prime[ # ]], 1] == 0 & ]
Select[Range[120], DigitCount[Prime[#], 10, 1]==0&] (* Harvey P. Dale, Jun 20 2023 *)
Numbers n such that 2^n-1 has only one primitive prime factor, sorted according to the magnitude of the corresponding prime.
+10
2
2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, 26, 42, 13, 34, 40, 32, 54, 17, 38, 27, 19, 33, 46, 56, 90, 78, 62, 31, 80
COMMENTS
Periods associated with A144755 in base 2. The binary analog of A051627.
EXAMPLE
2^12 - 1 = 4095 = 3 * 3 * 5 * 7 * 13, but none of 3, 5, 7 is a primitive prime factor, so the only primitive prime factor of 2^12 - 1 is 13.
MATHEMATICA
nmax = 65536; primesPeriods = Reap[Do[p = Cyclotomic[n, 2]/GCD[n, Cyclotomic[n, 2]]; If[PrimeQ[p], Print[n]; Sow[{p, n}]], {n, 1, nmax}]][[2, 1]]; Sort[primesPeriods][[All, 2]]
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