A014664 -id:A014664 - cp's OEIS frontend

A372801 Order of 16 modulo the n-th prime: least k such that prime(n) divides 16^k-1.

1, 1, 3, 5, 3, 2, 9, 11, 7, 5, 9, 5, 7, 23, 13, 29, 15, 33, 35, 9, 39, 41, 11, 12, 25, 51, 53, 9, 7, 7, 65, 17, 69, 37, 15, 13, 81, 83, 43, 89, 45, 95, 24, 49, 99, 105, 37, 113, 19, 29, 119, 6, 25, 4, 131, 67, 135, 23, 35, 47, 73, 51, 155, 39, 79, 15, 21, 173, 87, 22, 179

Offset: 2

Jianing Song, May 13 2024

a(n) is the period of the expansion of 1/prime(n) in hexadecimal.

Jianing Song, Table of n, a(n) for n = 2..10000

Cf. A302141 (order of 16 mod 2n+1).

In other bases: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.

PARI
```
a(n) = znorder(Mod(16, prime(n))).
```

a(n) = A014664(n)/gcd(4, A014664(n)) = A082654(n)/gcd(2, A082654(n)).

a(n) <= (prime(n) - 1)/2.

A201912 Irregular triangle of 2^k mod prime(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12, 1, 2, 4, 8, 16, 3, 6, 12, 24

Offset: 1

T. D. Noe, Dec 17 2011

The row lengths are in A014664. For n > 1, the first term of each row is 1 and the last term is 2*prime(n)-1, which is A006254. Many sequences are in this one.

			The first 11 rows are:
2:  0;
3:  1, 2;
5:  1, 2, 4, 3;
7:  1, 2, 4;
11: 1, 2, 4, 8,  5, 10,  9,  7,  3,  6;
13: 1, 2, 4, 8,  3,  6, 12, 11,  9,  5, 10,  7;
17: 1, 2, 4, 8, 16, 15, 13,  9;
19: 1, 2, 4, 8, 16, 13,  7, 14,  9, 18, 17, 15, 11,  3,  6, 12,  5, 10;
23: 1, 2, 4, 8, 16,  9, 18, 13,  3,  6, 12;
29: 1, 2, 4, 8, 16,  3,  6, 12, 24, 19,  9, 18,  7, 14, 28, 27, 25, 21, 13, 26, 23, 17, 5, 10, 20, 11, 22, 15;
31: 1, 2, 4, 8, 16;

T. D. Noe, Rows n = 1..60, flattened

Cf. A062117, A201908, A201909, A201910, A201911.

Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), A269266 (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67), A036135 (p=83), A036138 (p=101), A036140 (p=107), A036144 (p=131), A036146 (p=139), A036147 (p=149), A036150 (p=163), A036152 (p=173), A036153 (p=179), A036154 (p=181), A036157 (p=197), A036159 (p=211), A036161 (p=227).

P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(2,P[n]));;
Flat(Concatenation([0],List([2..10],n->List([0..R[n]-1],k->PowerMod(2,k,P[n]))))); # Muniru A Asiru, Feb 01 2019

nn = 10; p = 2; t = p^Range[0,Prime[nn]]; Flatten[Table[If[Mod[n, p] == 0, {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, Prime[Range[nn]]}]]

A226216 Primes p such that (p-1)/ord(2,p) > (q-1)/ord(2,q) for odd primes q < p.

Original entry on oeis.org

3, 7, 31, 73, 127, 601, 683, 1103, 1801, 2731, 5419, 8191, 43691, 61681, 65537, 121369, 122921, 131071, 178481, 262657, 524287, 2099863, 2796203, 6700417, 10567201, 13264529, 20394401, 48544121, 97685839, 112901153, 160465489, 164511353, 420778751, 536903681, 616318177

Offset: 1

Charles R Greathouse IV, May 31 2013

nonn

Essentially records in A001917. Sequence is infinite.

The Mersenne primes (A000668) are a subset. - Jeppe Stig Nielsen, Aug 30 2015

Amiram Eldar, Table of n, a(n) for n = 1..47

Cf. A000668, A001917, A014664, A152597, A152598.

r=0; p=3; L={}; While[Length@L < 20, v = (p-1)/ MultiplicativeOrder[2, p]; If[v > r, r = v; AppendTo[L, p]]; p = NextPrime@ p]; L (* Giovanni Resta, Aug 31 2015 *)

r=0;forprime(p=3,1e9,t=(p-1)/znorder(Mod(2,p));if(t>r,r=t;print1(p", ")))

a(n) = prime(A152597(n)). - Amiram Eldar, Nov 16 2023

A278966 Least Hamming weight of multiples of the n-th prime.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 2, 4, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 4, 2, 2, 4, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 4, 2, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 2, 2, 2

Offset: 1

Charles R Greathouse IV, Dec 02 2016

nonn

Since all primes after the first are odd, a(n) > 1 for n > 1.
a(n) = 2 if and only if A014664(n) is even, or equivalently prime(n) is not in A014663. - Robert Israel, Dec 08 2016
If prime(n) = A000668(k), then a(n) = A000043(k). - Robert Israel, Dec 20 2016

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A014663, A014664, A278967, A278968, A086342.

f:= proc(n) local p, R, V, W, k,v,r;
    p:= ithprime(n);
    R:= {seq(2 &^ i mod p, i=0..numtheory:-order(2,p)-1)};
    Rm:= map(t -> p-t, R);
    V:= R;
    W:= V;
    for k from 2 do
      if nops(V intersect Rm) > 0 then return k fi;
      V:= {seq(seq(v+r mod p, v=V),r=R)} minus W;
    W:= W union V;
    od
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Dec 20 2016

a[n_] := Module[{p, R, V, W, k, v, r}, p = Prime[n]; R = Union @ Table[ PowerMod[2, i, p], {i, 0, MultiplicativeOrder[2, p]-1}]; Rm = p - R; V = R; W = V; For[k = 2, True, k++, If[Length[V ~Intersection~ Rm] > 0, Return[k]]; V = Union@ Flatten@ Table[Table[v + Mod[r, p], {v, V}], {r, R}] ~Complement~ W; {W, W ~Union~ V}]];
a[1] = 1;
Array[a, 100] (* Jean-François Alcover, Jun 08 2020, after Robert Israel *)

a(n,p=prime(n))=my(o=znorder(Mod(2,p)), v1=Set(powers(Mod(2,p),o)), v=v1, s=1); while(!setsearch(v,Mod(0,p)), v=setbinop((x,y)->x+y,v,v1); s++); s

a(n) = A000120(A278967(n)). In particular, a(n) = A000120(prime(n)) whenever prime(n) is in A143027. - Max Alekseyev, May 22 2025

A245486 Product of the greatest prime factor of n and the greatest prime factor of n+1.

Original entry on oeis.org

2, 6, 6, 10, 15, 21, 14, 6, 15, 55, 33, 39, 91, 35, 10, 34, 51, 57, 95, 35, 77, 253, 69, 15, 65, 39, 21, 203, 145, 155, 62, 22, 187, 119, 21, 111, 703, 247, 65, 205, 287, 301, 473, 55, 115, 1081, 141, 21, 35, 85, 221, 689, 159, 33, 77, 133, 551, 1711, 295, 305

Offset: 1

Franklin T. Adams-Watters, Jul 23 2014

We take gpf(1) = 1 by convention.
Except for the initial 2, every member is in A006881.
2^n+1 is never divisible by 23, and when 2^n-1 is divisible by 23, it's also divisible by 89. So 46 cannot occur in the sequence. - Jack Brennen, Jul 23 2014
More generally, let m = A014664(i), i >= 2.  If m is odd, 2*A000040(i) occurs in the sequence iff A000040(i) = A006530(2^m-1), in which case it is a(2^m-1).  If m is even, 2*A000040(i) occurs in the sequence iff A000040(i) = A006530(2^(m/2)+1), in which case it is a(2^m). - Robert Israel, Jul 24 2014
If a(n) = prime(i)*prime(j), where i < j, then n <= A002072(j).  Using this, it can be shown that 3*89 does not occur in the sequence. - Robert Israel, Jul 24 2014
This sequence has an infinite limit; equivalently, each value in A006881 occurs only finitely many times in it. See A002072 for references.

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

Cf. A000040, A006530, A006881, A002072, A014664.

gpf:= n -> max(numtheory:-factorset(n)):
gpf(1):= 1:
seq(gpf(n)*gpf(n+1),n=1..100); # Robert Israel, Jul 24 2014

gpf[n_] := FactorInteger[n][[-1, 1]]; f[n_] := gpf[n] gpf[n + 1]; Array[f, 60] (* Robert G. Wilson v, Jul 23 2014 *)
Times@@@Partition[Table[FactorInteger[n][[-1,1]],{n,100}],2,1] (* Harvey P. Dale, Sep 24 2017 *)

gpf(n)=my(ps);if(n<=1,n,ps=factor(n)[,1]~;ps[#ps])
a(n) = gpf(n)*gpf(n+1)

a(n) = A006530(n) * A006530(n+1).

A263647 Numbers k such that 2^k-1 and 3^k-1 are coprime.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 13, 14, 15, 17, 19, 21, 25, 26, 27, 29, 31, 34, 37, 38, 39, 41, 45, 47, 49, 51, 53, 57, 59, 61, 62, 63, 65, 67, 71, 73, 74, 79, 81, 85, 87, 89, 91, 93, 94, 97, 98, 101, 103, 107, 109, 111, 113, 118, 122, 123, 125, 127, 133, 134, 135, 137, 139, 141, 142, 145, 147, 149, 151, 153, 157, 158, 159, 163, 167, 169, 171

Offset: 1

Robert Israel, Oct 22 2015

n such that there is no k for which both A014664(k) and A062117(k) divide n.
If n is in the sequence, then so is every divisor of n.
1 and 2 are the only members that are in A006093.
Conjectured to be infinite: see the Ailon and Rudnick paper.

			gcd(2^1-1, 3^1-1) = gcd(1,2) = 1, so a(1) = 1.
gcd(2^2-1, 3^2-1) = gcd(3,8) = 1, so a(2) = 2.
gcd(2^4-1, 3^4-1) = gcd(15,80) = 5, so 4 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Nir Ailon, Zéev Rudnick, Torsion points on curves and common divisors of a^k - 1 and b^k - 1, Acta Arith. 113 (2004), 31-38. Also arXiv:math/0202102 [math.NT], 2002.
Thomas Bloom, Problem 820, Erdős Problems.

Cf. A000225, A006093, A024023, A014664, A062117.

[n: n in [1..200] | Gcd(2^n-1, 3^n-1) eq 1]; // Vincenzo Librandi, May 01 2016

select(n -> igcd(2^n-1,3^n-1)=1, [$1..1000]);

Select[Range[200], GCD[2^# - 1, 3^# - 1] == 1 &] (* Vincenzo Librandi, May 01 2016 *)

A282902 Square array A(n, k) read by antidiagonals downwards: multiplicative order of 2 modulo prime(n)^k, where k runs over the positive integers.

Original entry on oeis.org

2, 6, 4, 18, 20, 3, 54, 100, 21, 10, 162, 500, 147, 110, 12, 486, 2500, 1029, 1210, 156, 8, 1458, 12500, 7203, 13310, 2028, 136, 18, 4374, 62500, 50421, 146410, 26364, 2312, 342, 11, 13122, 312500, 352947, 1610510, 342732, 39304, 6498, 253, 28, 39366, 1562500, 2470629, 17715610, 4455516, 668168, 123462, 5819, 812, 5

Offset: 2

Felix Fröhlich, Feb 24 2017

The number of initial terms in row n with constant values is equal to the highest value of x such that p = prime(n) satisfies 2^(p-1) == 1 (mod p^x).
From Robert Israel, Feb 24 2017: (Start)
a(n,k+1) is either a(n,k) or a(n,k)*prime(n). If it is a(n,k)*prime(n), then a(n,k+j) = a(n,k)*prime(n)^j for all j>=1.
a(n,2) = a(n,1) if and only if prime(n) is a Wieferich prime (A001220).
(End)

			Array A(n, k) starts
   2,   6,   18,     54,     162,      486,      1458
   4,  20,  100,    500,    2500,    12500,     62500
   3,  21,  147,   1029,    7203,    50421,    352947
  10, 110, 1210,  13310,  146410,  1610510,  17715610
  12, 156, 2028,  26364,  342732,  4455516,  57921708
   8, 136, 2312,  39304,  668168, 11358856, 193100552
  18, 342, 6498, 123462, 2345778, 44569782, 846825858

Robert Israel, Table of n, a(n) for n = 2..10012 (first 142 antidiagonals, flattened)

Cf. A014664 (column 1), A243905 (column 2).

Cf. A001220.

seq(seq(numtheory:-order(2,ithprime(i)^(m-i)),i=2..m-1),m=2..10); # Robert Israel, Feb 24 2017

A[n_, k_] := MultiplicativeOrder[2, Prime[n]^k];
Table[A[n-k+1, k], {n, 2, 11}, {k, n-1, 1, -1}] // Flatten (* Jean-François Alcover, Mar 02 2020 *)

a(n, k) = znorder(Mod(2, prime(n)^k))
array(rows, cols) = for(n=2, rows+1, for(k=1, cols, print1(a(n, k), ", ")); print(""))
array(7, 8) \\ print 7 X 8 array

A336719 Largest odd prime p for which the order of 2 mod p is at most n.

Original entry on oeis.org

3, 7, 7, 31, 31, 127, 127, 127, 127, 127, 127, 8191, 8191, 8191, 8191, 131071, 131071, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 524287, 2147483647, 2147483647, 2147483647, 2147483647, 2147483647, 2147483647

Offset: 2

Jeppe Stig Nielsen, Aug 01 2020

nonn

a(1) is undefined.
Changing "at most n" to "equal to n" in the definition gives A097406.
The first term that is not a Mersenne prime (A000668) is 4432676798593.
For a version without duplicates, see A336720. For a list of all n where a(n) increases, see A336721.

Jeppe Stig Nielsen, Table of n, a(n) for n = 2..251

Cf. A000668, A005420, A014664, A097406, A336720, A336721.

re=0;for(n=2,+oo,p=vecmax(factor(2^n-1)[,1]);p>re&&re=p;print1(re,", "))

A344202 Primes p such that gcd(ord_p(2), ord_p(3)) = 1.

Original entry on oeis.org

683, 599479, 108390409, 149817457, 666591179, 2000634731, 4562284561, 14764460089, 24040333283, 2506025630791, 5988931115977

Offset: 1

Sofia Lacerda, May 11 2021

'ord_p' here means the multiplicative order, not to be confused with the p-adic order that is also often denoted by ord_p.

Related to Diophantine equations of the form (2^x-1)*(3^y-1) = n*z^2.

Sofia Lacerda, C++ program (github).
MathOverflow, Solve this Diophantine equation (2^x-1)(3^y-1)=2z^2
Mathematics Stack Exchange, For all alpha,beta in N, are there only finitely many primes so that gcd(ordp(alpha),ordp(beta))=1?, 2021.
Wikipedia, Multiplicative order.
Wikipedia, P-adic_valuation.

Cf. A014664, A062117.

Select[Range[10^6], PrimeQ[#] && CoprimeQ[MultiplicativeOrder[2, #], MultiplicativeOrder[3, #]] &] (* Amiram Eldar, May 11 2021 *)

isok(p) = isprime(p) && (gcd(znorder(Mod(2, p)), znorder(Mod(3, p))) == 1); \\ Michel Marcus, May 11 2021

from sympy.ntheory import n_order
from sympy import gcd, nextprime
A344202_list, p = [], 5
while p < 10**9:
    if gcd(n_order(2,p),n_order(3,p)) == 1:
        A344202_list.append(p)
    p = nextprime(p) # Chai Wah Wu, May 12 2021

a(3)-a(5) from Michel Marcus, May 11 2021
a(6)-a(8) from Amiram Eldar, May 11 2021
a(9) from Daniel Suteu, May 16 2021
a(10) from Sofia Lacerda, Jul 07 2021
a(11) from Sofia Lacerda, Aug 03 2021

A085430 a(n) is the minimal m such that the group GL(m,2) has an element of order n.

Original entry on oeis.org

2, 2, 3, 4, 4, 3, 5, 6, 6, 10, 5, 12, 5, 4, 9, 8, 8, 18, 7, 5, 12, 11, 7, 20, 14

Offset: 2

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 18 2003

For n > 1, a(prime(n)) = A014664(n). Also, a(n) <= n. - Eric M. Schmidt, May 17 2013

Cf. A002884.

A085430 := function(n) local m; if IsPrime(n) and n>2 then return Order(2*Z(n)^0); fi; m := 1; while true do if ForAny(ConjugacyClasses(GL(m,2)), cc->Order(Representative(cc))=n) then return m; fi; m := m + 1; od; end; # Eric M. Schmidt, May 17 2013

Sequence extended and corrected by Eric M. Schmidt, May 17 2013

cp's OEIS Frontend

A372801 Order of 16 modulo the n-th prime: least k such that prime(n) divides 16^k-1.

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A201912 Irregular triangle of 2^k mod prime(n).

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A226216 Primes p such that (p-1)/ord(2,p) > (q-1)/ord(2,q) for odd primes q < p.

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A278966 Least Hamming weight of multiples of the n-th prime.

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Maple

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A245486 Product of the greatest prime factor of n and the greatest prime factor of n+1.

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Maple

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A263647 Numbers k such that 2^k-1 and 3^k-1 are coprime.

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Magma

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A282902 Square array A(n, k) read by antidiagonals downwards: multiplicative order of 2 modulo prime(n)^k, where k runs over the positive integers.

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Maple

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