A002371 -id:A002371 - cp's OEIS frontend

A060261 Denoting 5 consecutive primes by p, q, r, s and t, these are the values of q such that q, r and s have 10 as a primitive root, but p and t do not.

257, 379, 811, 971, 1097, 1217, 2411, 2539, 2617, 3011, 4051, 5297, 5657, 6211, 6337, 6659, 6857, 8647, 8807, 10457, 10651, 10687, 10937, 11731, 11939, 12451, 12577, 13099, 14011, 14537, 14731, 14887, 15137, 15607, 15737, 16091, 16411

Offset: 1

Jeff Burch, Mar 23 2001

nonn

A prime p has 10 as a primitive root iff the length of the period of the decimal expansion of 1/p is p-1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The indices of these primes are in A060260.

Cf. A001913, A002371, A060259, A060262.

test[p_] := MultiplicativeOrder[10, p]===p-1; Prime/@Select[Range[2, 2500], test[Prime[ # ]]&&test[Prime[ #+1]]&&test[Prime[ #+2]]&&!test[Prime[ #-1]]&&!test[Prime[ #+3]]&]

Edited by Dean Hickerson, Jun 17 2002

Offset corrected by Amiram Eldar, Oct 03 2021

A060262 a(n) is the smallest k such that prime(k), prime(k+1), ..., prime(k+n-1) all have 10 as a primitive root, but prime(k-1) and prime(k+n) do not.

Original entry on oeis.org

4, 17, 55, 7, 93, 754, 2611, 31092, 55207, 301252, 955428, 805428, 3651249, 3686621, 5510710, 42337888, 109670084, 590903433, 1010572448

Offset: 1

Jeff Burch, Mar 23 2001

A prime p has 10 as a primitive root iff the length of the period of the decimal expansion of 1/p is p-1.

a(21) = 9774718453 and a(23) = 9525468065. - Amiram Eldar, Oct 03 2021

Cf. A001913, A002371, A060259, A060260, A060261.

test[p_] := MultiplicativeOrder[10, p]===p-1; For[n=1, n<100, n++, a[n]=0]; v=4; While[True, For[n=1, test[Prime[v+n]], n++, Null]; If[a[n]==0, a[n]=v; Print["a(", n, ") = ", v]]; For[v+=n+1, !test[Prime[v]], v++, Null]]

Edited by Dean Hickerson, Jun 17 2002

a(13)-a(19) from Amiram Eldar, Oct 03 2021

A060282 Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's omitted).

Original entry on oeis.org

0, 3, 0, 142857, 9, 76923, 588235294117647, 52631578947368421, 434782608695652173913, 344827586206896551724137931, 32258064516129, 27, 2439, 23255813953488372093, 212765957446808510638297872340425531914893617

Offset: 1

N. J. A. Sloane, Mar 30 2001

			1/7 = 0.142857142..., so a(4) = 142857.
1/11 = 0.09090909..., so a(5) = 9.

T. D. Noe, Table of n, a(n) for n=1..100

Cf. A002371, A036275, A060283, A060251, A036275, A001913.

primePer[1] = primePer[3] = 0; primePer[n_] := FromDigits[(d = RealDigits[1/Prime[n]])[[1, 1]]] * 10^d[[2]]; Array[ primePer, 15] (* Amiram Eldar, Apr 28 2020 *)

f(n)=if(n<4,n==2,znorder(Mod(10, prime(n)))) \\ A002371
for(n=1,100,print1(floor(10^f(n)/prime(n)),","))

a(n) = floor(10^A002371(n)/prime(n)).
a(n) = 0 if and only if n = 1 or 3, corresponding to the primes 2 and 5, which are factors of 10. - Alonso del Arte, Apr 03 2020
ceiling(log_10(a(n))) = prime(n) - 1 if prime(n) is a full reptend prime (A001913). - Alonso del Arte, Apr 14 2020

More terms from Klaus Brockhaus, Mar 30 2001

A238104 Sum of digits in periodic part of decimal expansion of 1/prime(n).

Original entry on oeis.org

0, 3, 0, 27, 9, 27, 72, 81, 99, 126, 54, 9, 18, 90, 207, 63, 261, 270, 144, 126, 36, 54, 171, 198, 432, 18, 153, 225, 486, 504, 189, 585, 36, 207, 666, 306, 351, 360, 747, 207, 801, 810, 369, 864, 441, 405, 135, 999, 486, 1026, 1044, 18, 135, 225, 1152, 1179, 1206, 18, 324, 126, 621, 657, 675, 612, 1404, 351

Offset: 1

Kozhukhov Vlad, Dec 04 2013

Digit-sum of A060283(n).

			Prime(6) = 13, 1/13 = 0.076923076923076923076923..., the periodic part of which is 076923, whose digits add to 27 = a(6).
Since prime(n) must either divide or be coprime to 10, decimal expansions of prime(n) must either terminate or be purely recurrent, respectively. The only primes that divide 10 are prime(1) and prime(3), thus a(1) and a(3) = 0 as they have terminating decimal expansions. - _Michael De Vlieger_, May 20 2017

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A060283, A002371, A238105, A238106.

Table[Function[p, If[Divisible[10, p], 0, Total[RealDigits[1/p][[1, 1]]]]]@ Prime@ n, {n, 66}] (* Michael De Vlieger, May 20 2017 *)

forprime(i=1,1e2,print1(sumdigits((10^iferr(znorder(Mod(10,i)),E,0)-1)/i)", ")) \\ Lear Young, Mar 01 2014

a(n) = A007953(A060283(n)). - Michel Marcus, Mar 02 2014

Edited by David Applegate, Mar 01 2014

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

A246489 Duodecimal period of 1/(n-th prime) (0 by convention for the primes 2 and 3).

Original entry on oeis.org

0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280

Offset: 1

Eric Chen, Nov 15 2014

For p >= 5 (n >= 3): multiplicative order of 12 mod prime(n). - Joerg Arndt, Nov 15 2014

			For n=9, prime(9) = 23, 1/23 in base 12 is 0. 06316948421 06316948421 ..., which has period 11, so a(9) = 11.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002371 (decimal versions).

with(numtheory):
a:= n-> `if`(n<3, 0, order(12, ithprime(n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 16 2014

/* nonzero terms only: */
forprime(p=5,10^3,print1(znorder(Mod(12,p)),", ")); \\ Joerg Arndt, Nov 15 2014

a(n) = A246004(prime(n)).

A284601 Numbers k such that the decimal representation of 1/k does not terminate and has odd period.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 24, 27, 30, 31, 36, 37, 41, 43, 45, 48, 53, 54, 60, 62, 67, 71, 72, 74, 75, 79, 81, 82, 83, 86, 90, 93, 96, 106, 107, 108, 111, 120, 123, 124, 129, 134, 135, 142, 144, 148, 150, 151, 155, 158, 159, 162, 163, 164, 166, 172, 173, 180, 185, 186, 191, 192, 199, 201, 205, 212, 213, 214, 215

Offset: 1

Ilya Gutkovskiy, Mar 30 2017

From Robert G. Wilson v, Apr 02 2017: (Start)
If k is in the sequence, then so are 2k and 5k.
The complement of A284602.
Primitives: 3, 9, 27, 31, 37, 41, 43, 53, 67, 71, 79, 81, 83, 93, 107, 111, 123, ..., .
(End)
From Robert Israel, Apr 03 2017: (Start)
Numbers of the form 2^j * 5^k * m where m > 1, gcd(m,10)=1 and the multiplicative order of 10 (mod m) is odd.
Complement of A003592 in the multiplicative semigroup generated by A186635, i.e., numbers whose prime factors are in A186635 with at least one prime factor not 2 or 5. (End)

			27 is in the sequence because 1/27 = 0.0370(370)... period is 3, 3 is odd.
2 and 5 are not in the sequence because 1/2 = 0.5 and 1/5 = 0.2 are terminating expansions. See also comments in A051626 and A284602.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of 1/n

Cf. A002371, A003592, A003814, A051626, A186635, A284602.

filter:= proc(n) local m;
  m:= n/2^padic:-ordp(n,2);
  m:= m/5^padic:-ordp(m,5);
  m > 1 and numtheory:-order(10,m)::odd
end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 03 2017

Select[Range[215], Mod[Length[RealDigits[1/#][[1, -1]]], 2] == 1 & ]

A072858 Primes p such that the period of the decimal expansion of 1/p is a square.

Original entry on oeis.org

2, 3, 5, 17, 101, 163, 257, 577, 883, 1297, 1801, 3137, 3529, 5477, 7057, 7351, 8929, 9397, 11831, 12101, 13457, 13553, 14401, 15361, 15377, 15973, 19841, 20809, 21401, 21601, 23549, 24337, 25601, 29401, 30977, 33301, 33751, 33857, 38237

Offset: 1

Benoit Cloitre, Jul 26 2002

			The period of 1/17 = 0.05882352941176470588... is 16 = 4^2, hence 17 is in the sequence.
The period of 1/163 = 81 = 9^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002371.

Select[Prime[Range[4000]], IntegerQ @ Sqrt[Length[RealDigits[1/#][[1, 1]]]] &] (* Amiram Eldar, May 21 2022 *)

(a(n)=if(n<4,n==2,znorder(Mod(10, prime(n))))); for(n=1,1000,if(issquare(a(n))==1,print1(prime(n),","))) /* Thanks to Michael Somos for improvement of the PARI program */

A129727 Primes p for which the period length of 1/p is a semiprime.

Original entry on oeis.org

7, 13, 23, 31, 43, 47, 59, 67, 71, 101, 103, 139, 167, 179, 191, 263, 277, 283, 293, 311, 383, 431, 439, 443, 503, 547, 557, 599, 607, 613, 653, 683, 787, 809, 827, 853, 859, 863, 887, 947, 983, 997, 1013, 1019, 1039, 1163, 1213, 1237, 1321, 1367, 1399, 1423

Offset: 1

Jonathan Vos Post, May 12 2007

The prime index of A122183. Semiprime analog of A072859 = primes p for which the period length of 1/p is prime. Based upon A002371 = period of decimal expansion of 1/(n-th prime).

			a(1) = 7 because A000040(4) Period of decimal expansion of 1/7 = 6 = 2*3, a semiprime.
a(2) = 13 because A000040(6) = 6 = 2*3.
a(3) = 23 because A000040(9) = 22 = 2*11.
a(4) = 31 because A000040(11) = 15 = 3*5.
a(5) = 43 because A000040(14) = 21 = 3*7.
a(6) = 47 because A000040(15) = 46 = 2*23.
a(7) = 59 because A000040(17) = 58 = 2*29.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A002371, A072859, A128948.

fQ[p_] := Plus @@ Last /@ FactorInteger@Length@RealDigits[1/p][[1, 1]] == 2;; lst = {}; Do[ p = Prime@n; If[ fQ@p, AppendTo[lst, p]], {n, 230}] (* Robert G. Wilson v *)

A240665 Least k such that 10^k == -1 (mod prime(n)), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 3, 1, 3, 8, 9, 11, 14, 0, 0, 0, 0, 23, 0, 29, 30, 0, 0, 4, 0, 0, 22, 48, 2, 17, 0, 54, 56, 21, 65, 4, 23, 74, 0, 39, 0, 83, 0, 89, 90, 0, 96, 49, 0, 15, 111, 0, 114, 116, 0, 15, 25, 128, 131, 134, 0, 0, 14, 0, 73, 0, 0, 156, 0, 55, 168, 0, 58, 16, 0

Offset: 1

T. D. Noe, Apr 14 2014

nonn

The least k, if it exists, such that prime(n) divides 10^k + 1.

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

Cf. A002371 (order of 10 mod prime(n)), A068958.

Table[p = Prime[n]; s = Select[Range[p/2], PowerMod[10, #, p] == p - 1 &, 1]; If[s == {}, 0, s[[1]]], {n, 100}]

a(n) = A002371(n)/2 if A002371(n) is even, otherwise 0.

a(n) = A068958(n) for n > 3. - Georg Fischer, Oct 23 2018

cp's OEIS Frontend

A060261 Denoting 5 consecutive primes by p, q, r, s and t, these are the values of q such that q, r and s have 10 as a primitive root, but p and t do not.

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A060262 a(n) is the smallest k such that prime(k), prime(k+1), ..., prime(k+n-1) all have 10 as a primitive root, but prime(k-1) and prime(k+n) do not.

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A060282 Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's omitted).

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PARI

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A238104 Sum of digits in periodic part of decimal expansion of 1/prime(n).

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

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A246489 Duodecimal period of 1/(n-th prime) (0 by convention for the primes 2 and 3).

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Maple

PARI

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A284601 Numbers k such that the decimal representation of 1/k does not terminate and has odd period.

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Maple

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A072858 Primes p such that the period of the decimal expansion of 1/p is a square.