A024049 -id:A024049 - cp's OEIS frontend

A198768 a(n) = 9*5^n-1.

8, 44, 224, 1124, 5624, 28124, 140624, 703124, 3515624, 17578124, 87890624, 439453124, 2197265624, 10986328124, 54931640624, 274658203124, 1373291015624, 6866455078124, 34332275390624, 171661376953124, 858306884765624

Offset: 0

Vincenzo Librandi, Oct 30 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A024049, A057651, A081655.

Magma
```
[9*5^n-1: n in [0..30]];
```

9 5^Range[0, 20] - 1 (* or *) LinearRecurrence[{6, -5}, {8, 44}, 20] (* Harvey P. Dale, Nov 01 2011 *)
CoefficientList[Series[(8 - 4 x)/(1 - 6 x + 5 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 03 2013 *)

a(n) = 5*a(n-1)+4.
a(n) = 6*a(n-1)-5*a(n-2), n > 1.
G.f.: (8 - 4*x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 03 2013

A198769 a(n) = (9*5^n-1)/4.

Original entry on oeis.org

2, 11, 56, 281, 1406, 7031, 35156, 175781, 878906, 4394531, 21972656, 109863281, 549316406, 2746582031, 13732910156, 68664550781, 343322753906, 1716613769531, 8583068847656, 42915344238281, 214576721191406, 1072883605957031, 5364418029785156, 26822090148925781

Offset: 0

Vincenzo Librandi, Oct 30 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A024049, A057651, A081655.

Magma
```
[(9*5^n-1)/4: n in [0..30]];
```

(9*5^Range[0,30]-1)/4 (* or *) LinearRecurrence[{6,-5},{2,11},30] (* Harvey P. Dale, May 07 2012 *)

a(n) = 5*a(n-1)+1.
a(n) = 6*a(n-1)-5*a(n-2), n>1.
G.f.: (2-x)/(5*x^2-6*x+1). - Harvey P. Dale, May 07 2012

A198770 11*5^n-1.

Original entry on oeis.org

10, 54, 274, 1374, 6874, 34374, 171874, 859374, 4296874, 21484374, 107421874, 537109374, 2685546874, 13427734374, 67138671874, 335693359374, 1678466796874, 8392333984374, 41961669921874, 209808349609374, 1049041748046874

Offset: 0

Vincenzo Librandi, Oct 30 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A024049, A057651, A081655.

Magma
```
[11*5^n-1: n in [0..30]];
```

CoefficientList[Series[2*(5 - 3*x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
11*5^Range[0,20]-1 (* or *) LinearRecurrence[{6,-5},{10,54},30] (* Harvey P. Dale, May 14 2019 *)

a(n) = 5*a(n-1)+4.
a(n)= = 6*a(n-1)-5*a(n-2), n>1.
G.f.: 2*(5 - 3*x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013

A225586 Floor((5^n-1)/n).

Original entry on oeis.org

4, 12, 41, 156, 624, 2604, 11160, 48828, 217013, 976562, 4438920, 20345052, 93900240, 435965401, 2034505208, 9536743164, 44878791360, 211927625868, 1003867701480, 4768371582031, 22706531343005, 108372081409801, 518301258916440, 2483526865641276

Offset: 1

Vincenzo Librandi, May 30 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A024049, A034474, A082482, A129795.

Magma
```
[Floor((5^n-1)/n): n in [1..30]];
```

Table[Floor[(5^n - 1) / n], {n, 30}] (* or *) Table[Quotient[5^n - 1, n], {n, 30}]

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

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A198768 a(n) = 9*5^n-1.

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A198769 a(n) = (9*5^n-1)/4.

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A198770 11*5^n-1.

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A225586 Floor((5^n-1)/n).

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

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PARI

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

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Comments

Examples

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Programs

Magma

Mathematica