A019337 -id:A019337 - cp's OEIS frontend

A211243 Order of 7 mod n-th prime: least k such that prime(n) divides 7^k-1.

1, 1, 4, 0, 10, 12, 16, 3, 22, 7, 15, 9, 40, 6, 23, 26, 29, 60, 66, 70, 24, 78, 41, 88, 96, 100, 51, 106, 27, 14, 126, 65, 68, 69, 74, 150, 52, 162, 83, 172, 178, 12, 10, 24, 98, 99, 210, 37, 113, 228, 116, 238, 240, 125, 256, 262, 268, 135, 138, 20, 141, 292

Offset: 1

T. D. Noe, Apr 11 2012

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

Cf. A019337 (full reptend primes in base 7).
In other bases: A014664, A062117, A082654, A211241, A211242, A211244, A211245, A002371.
Row lengths of A201911. - Michel Marcus, Feb 04 2019

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(7,A000040[n])); # Muniru A Asiru, Feb 06 2019

nn = 7; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=7}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

A167795 Numbers with primitive root 7.

Original entry on oeis.org

2, 4, 5, 10, 11, 13, 17, 22, 23, 26, 34, 41, 46, 61, 67, 71, 79, 82, 89, 97, 101, 107, 121, 122, 127, 134, 142, 151, 158, 163, 169, 173, 178, 179, 194, 202, 211, 214, 229, 239, 241, 242, 254, 257, 263, 269, 289, 293, 302, 326, 338, 346, 347, 349, 358, 359, 379

Offset: 1

T. D. Noe, Nov 12 2009

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A019337 (primes with primitive root 7)

pr=7; Select[Range[2,2000], MultiplicativeOrder[pr,# ] == EulerPhi[ # ] &]

is(n)=if(n%7==0, return(0)); my(p=eulerphi(n)); znorder(Mod(7, n), p)==p \\ Charles R Greathouse IV, Jan 04 2025

A273948 Odd prime factors of generalized Fermat numbers of the form 7^(2^m) + 1 with m >= 0.

Original entry on oeis.org

5, 17, 257, 353, 769, 1201, 12289, 13313, 35969, 65537, 114689, 163841, 169553, 7699649, 9379841, 11886593, 28667393, 64749569, 70254593, 134818753, 197231873, 4643094529, 19847446529, 47072139617, 206158430209, 452850614273, 531968664833, 943558259713

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Odd primes p other than 3 such that the multiplicative order of 7 (mod p) is a power of 2.
From Robert Israel, Jun 16 2016: (Start)
If p is in the sequence, then for each m either p | 7^(2^k)+1 for some k < m or 2^m | p-1.  Thus all members except 5, 17, 353, 1201, 169553, 7699649, 134818753, 47072139617 are congruent to 1 mod 2^7.
The intersection of this sequence and A019337 is A019434 minus {3}. (End)

Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..34
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, pp. 397-405.
OEIS Wiki, Generalized Fermat numbers

Cf. A023394, A072982, A078304, A273945 (base 3), A273946 (base 5), A273947 (base 6), A273949 (base 11), A273950 (base 12).

filter:= proc(t)
  if not isprime(t) then return false fi;
  7 &^ (2^padic:-ordp(t-1,2)) mod t = 1
end proc:
select(filter, [seq(i,i=5..10^6,2)]); # Robert Israel, Jun 16 2016

Select[Prime@Range[3, 10^5], IntegerQ@Log[2, MultiplicativeOrder[7, #]] &]

A241045 Primes having primitive roots 2, 3, 5, and 7.

Original entry on oeis.org

173, 293, 677, 773, 797, 907, 1277, 1637, 1747, 2083, 2357, 2477, 2693, 2957, 3533, 3797, 4133, 4157, 4373, 4493, 4603, 4637, 4877, 4973, 5333, 5477, 5717, 5813, 5923, 6053, 6173, 6317, 6547, 6653, 6763, 7013, 7517, 8237, 8573, 8693, 8837, 9173, 9533

Offset: 1

T. D. Noe, Apr 16 2014

nonn

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

Cf. A001122, A019334, A019335, A019337, A213052.

fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[1200]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] &]
Select[Prime[Range[1200]],SubsetQ[PrimitiveRootList[#],{2,3,5,7}]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 16 2020 *)

A241047 Primes having primitive roots 2, 3, 5, 7, 11, and 13.

Original entry on oeis.org

293, 2477, 4373, 6173, 7013, 9173, 9677, 10853, 13037, 13397, 13613, 13877, 14957, 15413, 17093, 17597, 18413, 18917, 19157, 22277, 22613, 24317, 26813, 27653, 27893, 29333, 30197, 31517, 33893, 34613, 34877, 35573, 37253, 40493, 41117, 41333, 42437

Offset: 1

T. D. Noe, Apr 16 2014

nonn

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

Cf. A001122, A019334, A019335, A019337, A019339, A019341, A213052.

fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[4500]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] && fQ[13, #] &]

A241048 Primes having primitive roots 2, 3, 5, 7, 11, 13, and 17.

Original entry on oeis.org

2477, 9173, 10853, 13877, 14957, 15413, 22277, 22613, 24317, 27653, 30197, 34877, 37253, 41117, 41333, 42437, 42677, 43973, 48677, 51413, 55733, 61613, 62597, 63773, 66293, 72533, 73757, 74093, 76733, 79397, 79757, 82997, 86357, 90173, 92237, 92333, 95597

Offset: 1

T. D. Noe, Apr 16 2014

nonn

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

Cf. A001122, A019334, A019335, A019337, A019339, A019341, A019344, A213052.

fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[10000]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] && fQ[13, #] && fQ[17, #] &]

A071565 Numbers k such that x^k + x^(k-1) + x^(k-2) + ... + x + 1 is irreducible over GF(7).

Original entry on oeis.org

4, 10, 12, 16, 22, 40, 60, 66, 70, 78, 88, 96, 100, 106, 126, 150, 162, 172, 178, 210, 228, 238, 240, 256, 262, 268, 292, 346, 348, 358, 378, 396, 430, 432, 442, 460, 490, 498, 508, 520, 546, 576, 592, 598, 600, 630, 658, 676, 682, 732, 738, 742, 760, 772

Offset: 1

Robert G. Wilson v, Jun 22 2002

Cf. A071642.

Cf. A019337. - Joerg Arndt, Apr 17 2020

a(n) = A019337(n+1) - 1. - Joerg Arndt, Apr 17 2020

cp's OEIS Frontend

A211243 Order of 7 mod n-th prime: least k such that prime(n) divides 7^k-1.

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A167795 Numbers with primitive root 7.

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A273948 Odd prime factors of generalized Fermat numbers of the form 7^(2^m) + 1 with m >= 0.

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A241045 Primes having primitive roots 2, 3, 5, and 7.

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Mathematica

A241047 Primes having primitive roots 2, 3, 5, 7, 11, and 13.

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Mathematica

A241048 Primes having primitive roots 2, 3, 5, 7, 11, 13, and 17.

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Mathematica

A071565 Numbers k such that x^k + x^(k-1) + x^(k-2) + ... + x + 1 is irreducible over GF(7).

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