A028416 -id:A028416 - cp's OEIS frontend

A086998 Duplicate of A028416.

7, 11, 13, 17, 19, 23, 29, 47, 59, 61, 73, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139

Offset: 1

dead

A020806 Decimal expansion of 1/7.

1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2

Offset: 0

N. J. A. Sloane

142857 and 999999 = 7*142857 are first and last Kaprekar numbers with six digits. Note a(n) + a(n+3) = 9. (142857^2 = 20408122449; 20408 + 122449 = 142857.) a(n)^2 = 1, 16, 4, 64, 25, 49, ... - Paul Curtz, Aug 24 2009
The constant 19 + 1/7 = 19.142857... is the Kirchhoff index of the Möbius ladder graph on v=8 vertices. The Laplacian matrix has the eigenvalues 4 (one time), 4-sqrt(2) (2 times), 4+sqrt(2) (2 times), 2 (2 times) and 0 (one time). Then the Kirchhoff index is v times the sum over the inverse, nonzero eigenvalues. - R. J. Mathar, Feb 13 2011
Decimal expansion of -99*(zeta(-5) + zeta(-9)) - 1. - Arkadiusz Wesolowski, Sep 15 2013
Also, decimal expansion of Sum_{i>0} 1/8^i. - Bruno Berselli, Jan 03 2014
The points whose coordinates are overlapping pairs of digits of this sequence, (1, 4), (4, 2), (2, 8), (8, 5), (5, 7) and (7, 1), all lie on one ellipse, with equation 19*x^2 + 36*x*y + 41*y^2 - 333*x - 531*y = -1638. Overlapping pairs of pairs of digits, (14, 28), (42, 85), (28, 57), (85, 71), (57, 14), (71, 42), also yield 6 points on one ellipse, with equation -165104*x^2 + 160804*x*y + 8385498*x - 41651*y^2 - 3836349*y = 7999600. (See book by Wells and MathWorld link.) - M. F. Hasler, Oct 25 2017

			0.142857142857142857...

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrüche'.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.

J. Hall, One-Seventh Ellipse, on MathWorld, by E. W. Weisstein.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A002371, A028416, A049084, A068028.

I:=[1,4,2,8]; [n le 4 select I[n] else Self(n-1)-Self(n-3)+Self(n-4): n in [1..100]]; // Vincenzo Librandi, Mar 27 2015

Digits:=100: evalf(1/7); # Wesley Ivan Hurt, Jun 28 2016

CoefficientList[Series[(1 + 3 x - 2 x^2 + 7 x^3) / ((1 - x) (1 + x) (1 - x + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 27 2015 *)
realDigitsRecip[7] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Sep 18 2024 *)

1/7. \\ Charles R Greathouse IV, Sep 24 2015

digits(10^99\7) \\ M. F. Hasler, Oct 25 2017

From  Reinhard Zumkeller, Oct 06 2008: (Start)
A028416(1)=7; A002371(A049084(7)) = A002371(4) = 6.
a(n+6) = a(n), a(n+6/2) = 9 - a(n). (End)
From Colin Barker, Aug 14 2012: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
G.f.: (1+3*x-2*x^2+7*x^3) / ((1-x)*(1+x)*(1-x+x^2)). (End)
a(n) = A068028(n+2). - Zak Seidov, Mar 26 2015
a(n) = (27 - 11*cos(n*Pi) - 10*cos(n*Pi/3) - 6*sqrt(3)*sin(n*Pi/3))/6. - Wesley Ivan Hurt, Jun 28 2016
E.g.f.: (8*cosh(x) - exp(x/2)*(5*cos(sqrt(3)*x/2) + 3*sqrt(3)*sin(sqrt(3)*x/2)) + 19*sinh(x))/3. - Stefano Spezia, Dec 07 2024

A007450 Decimal expansion of 1/17.

Original entry on oeis.org

0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5

Offset: 0

N. J. A. Sloane

Period 16: repeat [0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7]. - Joerg Arndt, Mar 25 2013

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrueche'. [From Reinhard Zumkeller, Oct 06 2008]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
R. K. Hoeflin, Mega Test
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,-1,1).

I:=[0, 5, 8, 8, 2, 3, 5, 2, 9]; [n le 9 select I[n] else Self(n-1)-Self(n-8)+Self(n-9): n in [1..100]]; // Vincenzo Librandi, Mar 25 2013

CoefficientList[Series[-x (7 x^7 - 3 x^6 + 2 x^5 + x^4 - 6 x^3 + 3 x + 5)/((x - 1) (x^8+1)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 25 2013 *)

a(n)=[0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7][n%16+1]; /* Joerg Arndt, Mar 25 2013 */

From Reinhard Zumkeller, Oct 06 2008: (Start)
A028416(4)=17; A002371(A049084(17)) = A002371(7)=16;
a(n+16) = a(n), a(n+16/2) = 9 - a(n). (End)
G.f.: -x*(7*x^7-3*x^6+2*x^5+x^4-6*x^3+3*x+5)/((x-1)*(x^8+1)). - Colin Barker, Aug 15 2012

A087000 Half length of periodic part of decimal expansion of 1/p for those primes having a periodic part of even length.

Original entry on oeis.org

3, 1, 3, 8, 9, 11, 14, 23, 29, 30, 4, 22, 48, 2, 17, 54, 56, 21, 65, 4, 23, 74, 39, 83, 89, 90, 96, 49, 15, 111, 114, 116, 15, 25, 128, 131, 134, 14, 73, 156, 55, 168, 58, 16, 183, 93, 189, 191, 194, 100, 102, 209, 70, 216, 16, 76, 230, 77, 243, 245, 249, 251

Offset: 1

Reinhard Zumkeller, Jul 29 2003

a(n) appears to be the least k such that 10^k+1 is divisible by A028416(n). See A001271. - Michel Marcus, Aug 13 2023

Cf. A028416, A002371, A086999, A087001, A087002, A001271, A062397 (10^n+1).

a(n) = A002371(A049084(A028416(n)))/2.
a(n) = A055642(A086999(n))/2.
a(n) = A055642(A087001(n)) = A055642(A087002(n)).

A001271 Irregular table read by rows: row n lists prime factors of 10^n +1, with multiplicity.

Original entry on oeis.org

2, 11, 101, 7, 11, 13, 73, 137, 11, 9091, 101, 9901, 11, 909091, 17, 5882353, 7, 11, 13, 19, 52579, 101, 3541, 27961, 11, 11, 23, 4093, 8779, 73, 137, 99990001, 11, 859, 1058313049, 29, 101, 281, 121499449, 7, 11, 13, 211, 241, 2161, 9091, 353, 449, 641, 1409, 69857

Offset: 0

N. J. A. Sloane, revised Jul 13 2009

Except for 2, all these primes are in A028416, and all the primes in A028416 appear here. - Davide Rotondo, Aug 12 2023

			Table begins:
    2;
   11;
  101,
    7,      11, 13;
   73,     137;
   11,    9091;
  101,    9901;
   11,  909091;
   17, 5882353;
  ...

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Ray Chandler, Rows n=0..310, flattened (from Kamada link)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Makoto Kamada, Factorizations of 100...001.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A028416, A057934, A062397.

Term ordering corrected by Sean A. Irvine, Apr 11 2012

Minor edits to description by Ray Chandler, May 02 2017

A043292 Numbers that divide 10^k + 1 for some k.

Original entry on oeis.org

1, 2, 7, 11, 13, 17, 19, 23, 29, 47, 49, 59, 61, 73, 77, 89, 91, 97, 101, 103, 109, 113, 121, 127, 131, 133, 137, 139, 143, 149, 157, 161, 167, 169, 179, 181, 193, 197, 209, 211, 223, 229, 233, 241, 247, 251, 253, 257, 263, 269, 281, 289, 293, 299, 313, 329

Offset: 1

Vladeta Jovovic, Apr 04 2002

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014657, A069531.

Odd primes in this sequence are A028416.

filter:= n -> traperror(NumberTheory:-ModularLog(-1,10,n)) <> "no solutions exist":
filter(2):= true:
select(filter, [$1..500]); # Robert Israel, Feb 11 2023

A186635 Primes p such that the decimal expansion of 1/p has a periodic part of odd length.

Original entry on oeis.org

2, 3, 5, 31, 37, 41, 43, 53, 67, 71, 79, 83, 107, 151, 163, 173, 191, 199, 227, 239, 271, 277, 283, 307, 311, 317, 347, 359, 397, 431, 439, 443, 467, 479, 523, 547, 563, 587, 599, 613, 631, 643, 683, 719, 733, 751, 757, 773, 787, 797, 827, 839, 853, 883, 907, 911, 919, 947, 991, 1013, 1031, 1039, 1093, 1123, 1151, 1163, 1187

Offset: 1

Jani Melik, Feb 24 2011

Interestingly, the initial terms of A040119 (Primes p such that x^4 = 10 has a solution mod p) are identical to the initial terms of this sequence except for 241 which is a term of A040119 but not of A186635. [John W. Layman, Feb 25 2011]

There are many numbers in A040119 that are not here: 241, 641, 769, 809, 1009, 1409, 1601, 1721.... - T. D. Noe, Feb 25 2011

Cf. A002371, A048595, A028416 (complement in the primes), A040119.

Ax := proc(n) local st:
st := ithprime(n):
if (modp(numtheory[order](10,st),2) <> 0) then
   RETURN(st)
fi: end:  seq(Ax(n), n=1..200);

Union[{2, 5}, Select[Prime[Range[200]], OddQ[Length[RealDigits[1/#][[1, 1]]]] &]]

select( {is_A186635(n)=isprime(n) && (n<7 || znorder(Mod(10, n))%2)}, [0..1234]) \\ M. F. Hasler, Nov 19 2024

from sympy import isprime, n_order
is_A186635 = lambda n: isprime(n) and (n<7 or n_order(10, n)%2)
[n for n in range(1234) if is_A186635(n)] # M. F. Hasler, Nov 19 2024

A187040 Numbers for which Midy's theorem holds.

Original entry on oeis.org

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

Jani Melik, Mar 02 2011

Martin gives generalizations of Midy's theorem that characterize the numbers in this sequence. See theorem 8. - T. D. Noe, Mar 02 2011

García-Pulgarín Gilberto and Giraldo Hernán give the characterization of the numbers that satisfy Midy's property.

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

Cf. A028416, A187041.

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);

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 *)

Corrected by T. D. Noe, Mar 02 2011

A139886 Primes of the form 10x^2 + 19y^2.

Original entry on oeis.org

19, 29, 59, 109, 179, 181, 211, 269, 331, 379, 421, 509, 659, 661, 811, 829, 941, 971, 1019, 1021, 1091, 1171, 1181, 1229, 1291, 1381, 1459, 1549, 1571, 1579, 1699, 1709, 1741, 1789, 1861, 1931, 1979, 2029, 2131, 2141, 2179, 2269, 2309, 2339

Offset: 1

T. D. Noe, May 02 2008

Discriminant = -760. See A139827 for more information.
10*x^2 + 19 produces 19 consecutive primes belonging to A028416 for x from 0 to 18. - Davide Rotondo, Jun 13 2022
Primes p such that Kronecker(2,p) <= 0, Kronecker(5,p) >= 0 and Kronecker(-19,p) <= 0. - Jianing Song, Jun 13 2022

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Apart from 19, intersection of A003629, A045468 and A191063.

[ p: p in PrimesUpTo(3000) | p mod 760 in {19, 21, 29, 51, 59, 69, 91, 109, 141, 179, 181, 189, 211, 219, 221, 259, 261, 269, 299, 331, 341, 371, 379, 411, 421, 451, 459, 469, 509, 531, 611, 621, 629, 659, 661, 699, 749}]; // Vincenzo Librandi, Jul 30 2012

QuadPrimes2[10, 0, 19, 10000] (* see A106856 *)

The primes are congruent to {19, 21, 29, 51, 59, 69, 91, 109, 141, 179, 181, 189, 211, 219, 221, 259, 261, 269, 299, 331, 341, 371, 379, 411, 421, 451, 459, 469, 509, 531, 611, 621, 629, 659, 661, 699, 749} (mod 760). [For the other direction, primes satisfying this congruence are terms of this sequence since 760 is a term in A003171. - Jianing Song, Jun 13 2022]

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

Original entry on oeis.org

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

cp's OEIS Frontend

A086998 Duplicate of A028416.

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A020806 Decimal expansion of 1/7.

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A007450 Decimal expansion of 1/17.

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A087000 Half length of periodic part of decimal expansion of 1/p for those primes having a periodic part of even length.

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A001271 Irregular table read by rows: row n lists prime factors of 10^n +1, with multiplicity.

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A043292 Numbers that divide 10^k + 1 for some k.

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Maple

A186635 Primes p such that the decimal expansion of 1/p has a periodic part of odd length.

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A187040 Numbers for which Midy's theorem holds.

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