A024088 -id:A024088 - cp's OEIS frontend

A274908 Largest prime factor of 8^n - 1.

7, 7, 73, 13, 151, 73, 337, 241, 262657, 331, 599479, 109, 121369, 5419, 23311, 673, 131071, 262657, 1212847, 1321, 649657, 599479, 10052678938039, 38737, 10567201, 22366891, 97685839, 14449, 9857737155463, 18837001, 658812288653553079, 22253377

Offset: 1

Vincenzo Librandi, Jul 11 2016

nonn

			8^5 -1 = 32767 = 7*31*151, so a(5) = 151.

Table of n, a(n) for n = 1..500
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A005420, A006530, A024088, A274905.

Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(8^n-1)): n in [1..40]];

f:= n -> max(map(t -> max(numtheory:-factorset(subs(x=2,t[1]))), factors(x^(3*n)-1)[2])):
map(f, [$1..120]); # Robert Israel, Jul 12 2016

Table[FactorInteger[8^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024088(n)). - Michel Marcus, Jul 11 2016

a(n) = A005420(3*n). - Robert Israel, Jul 12 2016

Terms to a(100) in b-file from Vincenzo Librandi, Jul 13 2016
a(101)-a(402) in b-file from Amiram Eldar, Feb 02 2020
a(403)-a(500) in b-file from Max Alekseyev, Apr 25 2022, Sep 11 2022, Dec 05 2022, Feb 25 2023

A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)(2^(3n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

Original entry on oeis.org

1, 15, 219, 3315, 51491, 811395, 12882499, 205321155, 3278747331, 52408827075, 838132189379, 13406842675395, 214483303960771, 3431523432591555, 54902699475185859, 878429788032676035, 14054769379960303811, 224875452250864496835, 3598000373385828511939

Offset: 1

Peter Bala, Feb 28 2014

This is a fourth-order linear divisibility sequence, that is, the sequence satisfies a linear recurrence of order 4 and if n | m then a(n) | a(m). This is a consequence of the following more general result: The polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a fourth-order linear divisibility sequence in the polynomial ring Z[x,y]. See the Bala link.
Hence, for a fixed integers M and N, the normalized sequence (M^n + N^n)*(M^(3*n) - N^(3*n))/ ( (M + N)*(M^3 - N^3) ) for n = 1,2,3,... is a linear divisibility sequence of order 4. It has the rational o.g.f. x*(1 - 2*M*N*(M^2 - M*N + N^2)*x + (M*N)^4*x^2)/( (1 - M^4*x)*(1 - M^3*N*x)*(1 - M*N^3*x)*(1 - N^4*x) ). This is the case M = 2, N = 1. For other cases see A238539(M = 2, N = -1), A238540(M = 3, N = 1) and A238541(M = 3, N = 2). See also A238536, A238537 and A215466.
Note, these sequences do not belong to the family of linear divisibility sequences of the fourth order studied by Williams and Guy, which have o.g.f.s of the form x*(1 - q*x^2)/Q(x), Q(x) a quartic polynomial and q an integer parameter.

Michael De Vlieger, Table of n, a(n) for n = 1..831
Peter Bala, A family of linear divisibility sequences of order four
E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
Wikipedia, Divisibility sequence
Wikipedia, Lucas Sequence
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (27,-202,432,-256).

Cf. A215466, A236536, A236537, A238539, A238540, A238541.

seq(1/21*(2^n + 1)*(2^(3*n) - 1), n = 1..20);

LinearRecurrence[{27,-202,432,-256},{1,15,219,3315},20] (* Harvey P. Dale, Jul 04 2019 *)

a(n) = (1/21)*(2^n + 1)*(2^(3*n) - 1) = A000051(n)*A024088(n)/21.
a(n) = (1/21)*(4^n - 1)*(8^n - 1)/(2^n - 1).
O.g.f.: x*(1 - 12*x + 16*x^2)/((1 - x)*(1 - 2*x)*(1 - 8*x)*(1 - 16*x)).
Recurrence equation: a(n) = 27*a(n-1) - 202*a(n-2) + 432*a(n-4) - 256*a(n-4).

A125837 Numbers whose base 8 or octal representation is 6666666......6.

Original entry on oeis.org

0, 6, 54, 438, 3510, 28086, 224694, 1797558, 14380470, 115043766, 920350134, 7362801078, 58902408630, 471219269046, 3769754152374, 30158033218998, 241264265751990, 1930114126015926, 15440913008127414, 123527304065019318, 988218432520154550, 7905747460161236406

Offset: 1

Zerinvary Lajos, Feb 03 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A001018, A023001, A024088, A083068.

List([1..30], n-> 6*(8^(n-1)-1)/7); # G. C. Greubel, Aug 03 2019

[6*(8^(n-1)-1)/7: n in [1..30]]; // G. C. Greubel, Aug 03 2019

Maple
```
seq(6*(8^n-1)/7, n=0..30);
```

FromDigits[#,8]&/@Table[Table[6,{i}],{i,0,30}]  (* Harvey P. Dale, Mar 19 2011 *)
6*(8^(Range[30]-1) -1)/7 (* G. C. Greubel, Aug 03 2019 *)

vector(30, n, 6*(8^(n-1)-1)/7) \\ G. C. Greubel, Aug 03 2019

[6*(8^(n-1)-1)/7 for n in (1..30)] # G. C. Greubel, Aug 03 2019

a(n) = 6*(8^(n-1) -1)/7 = 6*A023001(n-1).
a(n) = 8*a(n-1) + 6 for n>1, a(1)=0. - Vincenzo Librandi, Oct 03 2010
G.f.: 6*x^2/( (1-x)*(1-8*x) ). - R. J. Mathar, Oct 07 2016
E.g.f.: 6*(exp(8*x) - exp(x))/7. - G. C. Greubel, Aug 03 2019
a(n) = -1 + A083068(n-1). - Alois P. Heinz, May 20 2023

A125836 Numbers whose base 8 or octal representation is 555555555......5.

Original entry on oeis.org

0, 5, 45, 365, 2925, 23405, 187245, 1497965, 11983725, 95869805, 766958445, 6135667565, 49085340525, 392682724205, 3141461793645, 25131694349165, 201053554793325, 1608428438346605, 12867427506772845, 102939420054182765

Offset: 1

Zerinvary Lajos, Feb 03 2007

			Octal...............decimal
0........................0
5........................5
55......................45
555....................365
5555..................2925
55555................23405
555555..............187245
5555555............1497965
55555555..........11983725
555555555.........95869805
5555555555.......766958445
etc. ...............etc.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A023001, A024088.

List([1..30], n-> 5*(8^(n-1) -1)/7); # G. C. Greubel, Aug 03 2019

[5*(8^(n-1) -1)/7: n in [1..30]]; // G. C. Greubel, Aug 03 2019

Maple
```
seq(5*(8^n-1)/7, n=0..30);
```

5*(8^(Range[30]-1) -1)/7 (* G. C. Greubel, Aug 03 2019 *)

vector(30, n, 5*(8^(n-1) -1)/7) \\ G. C. Greubel, Aug 03 2019

[5*(8^(n-1) -1)/7 for n in (1..30)] # G. C. Greubel, Aug 03 2019

a(n) = 5*(8^(n-1) -1)/7 = 5*A023001(n-1).
a(n) = 8*a(n-1) + 5, with a(1)=0. - Vincenzo Librandi, Sep 30 2010
G.f.: 5*x^2/( (1-x)*(1-8*x)). - R. J. Mathar, Sep 30 2013
From G. C. Greubel, Aug 03 2019: (Start)
a(n) = 5*A024088(n-1)/7.
E.g.f.: 5*(exp(8*x) - exp(x))/7. (End)

A167617 G.f.: x^2(3+3x+x^2) / ( (2x+1) (1+x) * (1+x+x^2) * (x^2-x+1) ) .

Original entry on oeis.org

0, 0, 3, -6, 10, -21, 42, -84, 171, -342, 682, -1365, 2730, -5460, 10923, -21846, 43690, -87381, 174762, -349524, 699051, -1398102, 2796202, -5592405, 11184810, -22369620, 44739243, -89478486, 178956970, -357913941, 715827882, -1431655764, 2863311531

Offset: 0

Paul Curtz, Nov 07 2009

The derived sequence a(n+1) + 2*a(n) reads 0,3,0,-2,-1,0 (and repeat with period 6).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3, -3, -3, -3, -3, -2).

Cf. A167613.

CoefficientList[Series[x^2(3+3x+x^2)/((2x+1)(1+x)(1+x+x^2)(x^2-x+1)), {x,0,40}],x] (* or *) LinearRecurrence[{-3,-3,-3,-3,-3,-2},{0,0,3,-6,10,-21},40] (* Harvey P. Dale, Sep 08 2011 *)

a(3*k+2) + a(3*k+3) + a(3*k+4) = (-1)^(k+1)*A024088(k+1).
a(n) = (-1)^n*A024495(n+1) + A131531(n+1).
a(n) = -3*a(n-1) -3*a(n-2) -3*a(n-3) -3*a(n-4) -3*a(n-5) -2*a(n-6).

Edited and extended by R. J. Mathar, Nov 12 2009

A333813 a(n) = 2^(1 + floor(n*log_2(3))) - (3^n + 1).

Original entry on oeis.org

0, 0, 6, 4, 46, 12, 294, 1908, 1630, 13084, 6486, 84996, 517134, 502828, 3605638, 2428308, 24062142, 5077564, 149450422, 985222180, 808182894, 6719515980, 2978678758, 43295774644, 267326277406, 252223018332, 1856180682774, 1170495537220

Offset: 0

Ctibor O. Zizka, Apr 06 2020

nonn

For integers X, Y, let a(n) = (X^(t+1) - 1) / (X - 1) - Y^n, where t = floor(n*log_X(Y)) . This sequence is for X = 2, Y = 3.

			a(0) = 2^(1 + floor(0*log_2(3))) - (3^0 + 1) = 0; a(4) = 2^(1 + floor(4*log_2(3))) - (3^4 + 1) = 46.

Cf. A000225, A024036.

Examples for integers X = Y from {2, 3, 4, 5, 6, 7, 8, 9, 10} are A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275. Examples for X = 2, Y = 4 are A024036; for X = 2, Y = 8, A024088; and for X = 3, Y = 9, A191681.

Table[2^(1+Floor[n Log2[3]])-(3^n+1),{n,0,30}] (* Harvey P. Dale, Sep 04 2023 *)

a(n) = 2^(1 + floor(n*log_2(3))) - (3^n + 1).

cp's OEIS Frontend

A274908 Largest prime factor of 8^n - 1.

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A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)*(2^(3*n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

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A125837 Numbers whose base 8 or octal representation is 6666666......6.

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A125836 Numbers whose base 8 or octal representation is 555555555......5.

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A167617 G.f.: x^2*(3+3*x+x^2) / ( (2*x+1) * (1+x) * (1+x+x^2) * (x^2-x+1) ) .

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Mathematica

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A333813 a(n) = 2^(1 + floor(n*log_2(3))) - (3^n + 1).

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Mathematica

Formula

A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)(2^(3n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

A167617 G.f.: x^2(3+3x+x^2) / ( (2x+1) (1+x) * (1+x+x^2) * (x^2-x+1) ) .