A260119 -id:A260119 - cp's OEIS frontend

A086892 Greatest common divisor of 2^n-1 and 3^n-1.

1, 1, 1, 5, 1, 7, 1, 5, 1, 11, 23, 455, 1, 1, 1, 85, 1, 133, 1, 275, 1, 23, 47, 455, 1, 1, 1, 145, 1, 2387, 1, 85, 23, 1, 71, 23350145, 1, 1, 1, 11275, 1, 2107, 431, 115, 1, 47, 1, 750295, 1, 11, 1, 265, 1, 133, 23, 145, 1, 59, 1, 47322275, 1, 1, 1, 85, 1, 10787, 1, 5, 47, 781, 1

Offset: 1

Joseph H. Silverman (jhs(AT)math.brown.edu), Sep 18 2003

a(n) is a simple (the simplest?) example of a divisibility sequence associated to a rational point on an algebraic group of dimension larger than two. Specifically, it is the divisibility sequence associated to the point (2,3) on the two-dimensional torus G_m^2. Ailon and Rudnick conjecture that a(n) = 1 for infinitely many n.

According to Corvaja, a(n) < 2^n - 1 for all but finitely many n.

Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of a^n-1 and b^n-1. Math. Z. 243 (2003), no. 1, 79-84

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
N. Ailon, Z. Rudnick, Torsion points on curves and common divisors of a^k-1 and b^k-1, Acta Arith. 113 (2004), no. 1, 31-38.
Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of a^n-1 and b^n-1, Math. Z. 243 (2003), no. 1, 79-84
P. Corvaja, Greatest Common Divisors in Vojta's Conjecture: Arithmetic and Geometry, Journées Arithmétiques 2011.
Index to divisibility sequences

Cf. A000225, A000255, A003462, A024023, A260119.

a086892 n = a086892_list !! (n-1)
a086892_list = tail $ zipWith gcd a000225_list a003462_list
-- Reinhard Zumkeller, Jul 18 2015

[Gcd(2^n-1, 3^n-1): n in [1..75]]; // Vincenzo Librandi, Sep 02 2015

seq(igcd(2^n-1,3^n-1), n=1..100); # Robert Israel, Sep 02 2015

Table[GCD[2^n - 1, 3^n - 1], {n, 100}] (* Vincenzo Librandi, Sep 02 2015 *)

PARI
```
vector(100,n,gcd(2^n-1,3^n-1))
```

a(n) = gcd(2^n - 1, 3^n - 1).

a(n) = GCD(A000255(n), A003462(n)) = GCD(A000255(n), A024023(n)). - Reinhard Zumkeller, Mar 26 2004

Replaced arXiv URL with non-cached version by R. J. Mathar, Oct 23 2009

A268081 Least positive integer k such that 3^n-1 and k^n-1 are relatively prime.

Original entry on oeis.org

2, 2, 2, 10, 2, 28, 2, 10, 2, 22, 10, 910, 2, 2, 2, 170, 2, 3458, 2, 110, 2, 46, 10, 910, 2, 2, 2, 290, 2, 9548, 2, 340, 10, 2, 22, 639730, 2, 2, 2, 4510, 2, 1204, 10, 230, 2, 94, 2, 216580, 2, 22, 2, 530, 2, 3458, 22, 580, 2, 118, 2, 18928910

Offset: 1

Tom Edgar, Jan 25 2016

nonn

Note that (3^n-1)^n-1 is always relatively prime to 3^n-1.
According to the conjecture given in A086892, a(n) = 2 infinitely often.
When n>1, a(n) = 2 if and only if A260119(n) = 3.
From Robert Israel, Nov 20 2024: (Start)
  a(n) <= a(m*n) for m >= 1.
  If p is a prime factor of 3^n - 1 such that p-1 divides n, then a(n) is a multiple of p. (End)

			Since 3^5-1 = 242 and 2^5-1 = 31 are relatively prime, a(5) = 2.

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

Cf. A086892, A260119.

f:= proc(n) local t,F,m,k,v;
       t:= 3^n-1;
       F:= select(isprime,map(`+`,numtheory:-divisors(n),1));
       m:= convert(select(s -> t mod s = 0, F),`*`);
       for k from m by m do
             v:= k &^ n - 1 mod t;
             if igcd(v, t) = 1 then return k fi
       od
    end proc:
map(f, [$1..100]); # Robert Israel, Nov 20 2024

Table[k = 1; While[! CoprimeQ[3^n - 1, k^n - 1], k++]; k, {n, 59}] (* Michael De Vlieger, Jan 27 2016 *)

a(n) = {k=1; while( gcd(3^n-1, k^n-1)!=1, k++); k; }

def min_k(n):
    g, k=2, 0
    while g!=1:
        k=k+1
        g=gcd(3^n-1, k^n-1)
    return k
print([min_k(n) for n in [1..60]])

A270360 Least positive integer k such that 5^n-1 and k^n-1 are relatively prime.

Original entry on oeis.org

2, 6, 2, 6, 2, 42, 2, 6, 2, 132, 2, 546, 2, 12, 6, 102, 2, 798, 2, 198, 2, 138, 2, 546, 2, 6, 2, 348, 2, 85932, 2, 102, 2, 12, 22, 383838, 2, 12, 6, 2706, 2, 1806, 2, 414, 22, 282, 2, 9282, 2, 264, 2, 318, 2, 1596, 2, 348, 2, 354, 2, 34072038

Offset: 1

Tom Edgar, Mar 16 2016

nonn

Note that (5^n-1)^n-1 is always relatively prime to 5^n-1.
Based on conjecture given in A270390, a(n) = 2 infinitely often.
Are all terms even? - Harvey P. Dale, Jul 29 2024

			Since 5^2-1 = 24 and 6^2-1 = 35 are relatively prime while 2^2-1, 3^2-1, 4^2-1, and 5^2-1 are not relatively prime to 24, a(5) = 3.

Cf. A260119, A268081.

lpi[n_]:=Module[{k=1,c=5^n-1},While[!CoprimeQ[c,k^n-1],k++];k]; Array[lpi,60] (* Harvey P. Dale, Jul 29 2024 *)

a(n) = {k=1; while( gcd(5^n-1, k^n-1)!=1, k++); k; }

def min_k(n):
    g, k=2, 0
    while g!=1:
        k=k+1
        g=gcd(5^n-1, k^n-1)
    return k
print([min_k(n) for n in [1..60]])

a(60) from Harvey P. Dale, Jul 29 2024

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A086892 Greatest common divisor of 2^n-1 and 3^n-1.

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A268081 Least positive integer k such that 3^n-1 and k^n-1 are relatively prime.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

A270360 Least positive integer k such that 5^n-1 and k^n-1 are relatively prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Sage

Extensions