A270390 -id:A270390 - cp's OEIS frontend

A349748 Primes p for which 2^p-1 and 5^p-1 are not relatively prime.

2, 179, 239, 359, 419, 431, 499, 547, 571, 641, 659, 719, 761, 937, 1013, 1019, 1223, 1439, 1499, 1559, 1789, 2039, 2339, 2399, 2459, 2539, 2593, 2677, 2699, 2819, 2939, 3299, 3359, 3539, 3779, 4013, 4019, 4273, 4513, 4787, 4919, 5039, 5279, 5393, 5399, 5639, 6173, 6199, 6899, 7079, 8599, 8741, 8929, 9059, 9419, 9479

Offset: 1

Antti Karttunen, Nov 30 2021

nonn

Primes p for which A270390(p) = gcd(A000225(p), A024049(p)) > 1.

			2 is included as 2^2 - 1 = 3 and 5^2 - 1 = 24 share a prime factor 3.

Cf. A000225, A024049, A270390.

upto=10^4;Select[Prime[Range[PrimePi[upto]]],GCD[2^#-1,5^#-1]>1&] (* Paolo Xausa, Nov 30 2021 *)

isA349748(n) = (isprime(n)&&(gcd(2^n-1,5^n-1)>1));

from math import gcd
from sympy import isprime
def ok(n): return isprime(n) and gcd(2**n-1, 5**n-1) > 1
print([k for k in range(9500) if ok(k)]) # Michael S. Branicky, Nov 30 2021

A265166 Numbers n such that 2^n-1 and 5^n-1 are coprime.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 37, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 97, 101, 103, 107, 109, 111, 113, 115, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 143

Offset: 1

Vincenzo Librandi, May 01 2016

nonn

Also numbers n such that A270390(n) = 1.

Conjectured to be infinite: see the Ailon and Rudnick paper.

			gcd(2^1-1, 5^1-1) = gcd(1,4) = 1, so a(1) = 1.
gcd(2^3-1, 5^3-1) = gcd(7,124) = 1, so a(2) = 3.
gcd(2^4-1, 5^4-1) = gcd(15,624) = 3, so 4 is not in the sequence.

N. Ailon and Z. Rudnick, Torsion points on curves and common divisors of a^k - 1 and b^k - 1 , Acta Arith. 113 (2004), pp. 31-38.

Cf. A000225, A024049, A263647, A270390.

[n: n in [1..200] | Gcd(2^n-1,5^n-1) eq 1];

Select[Range[200], GCD[2^# - 1, 5^# - 1] == 1 &]
Select[Range[150],CoprimeQ[2^#-1,5^#-1]&] (* Harvey P. Dale, Apr 12 2018 *)

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

cp's OEIS Frontend

A349748 Primes p for which 2^p-1 and 5^p-1 are not relatively prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

A265166 Numbers n such that 2^n-1 and 5^n-1 are coprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

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