A001018 -id:A001018 - cp's OEIS frontend

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

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

A199301 a(n) = (2n+1)*8^n.

Original entry on oeis.org

1, 24, 320, 3584, 36864, 360448, 3407872, 31457280, 285212672, 2550136832, 22548578304, 197568495616, 1717986918400, 14843406974976, 127543348822016, 1090715534753792, 9288674231451648, 78812993478983680, 666532744850833408, 5620492334958379008, 47269781688880726016

Offset: 0

Philippe Deléham, Nov 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001018 (Powers of 8), A005408 (2n+1).

Cf. A014480, A058962, A124647, A155988, A171220, A199299, A199300.

[(2*n+1)*8^n: n in [0..30]]; // Vincenzo Librandi, Nov 05 2011

A199301:=n->(2*n+1)*8^n: seq(A199301(n), n=0..20); # Wesley Ivan Hurt, Oct 30 2014

Table[(2 n + 1)*8^n, {n, 0, 20}] (* Wesley Ivan Hurt, Oct 30 2014 *)

a(n) = (2*n+1)*8^n \\ Amiram Eldar, Dec 10 2022

a(n) = 16*a(n-1)-64*a(n-2).
G.f.: (1+8*x)/(1-8*x)^2.
a(n) = 8*(a(n-1)+2^(3*n-2)). - Vincenzo Librandi, Nov 05 2011
a(n) = A005408(n) * A001018(n). - Wesley Ivan Hurt, Oct 30 2014
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(8)*arccoth(sqrt(8)).
Sum_{n>=0} (-1)^n/a(n) = sqrt(8)*arccot(sqrt(8)). (End)
E.g.f.: exp(8*x)*(1 + 16*x). - Stefano Spezia, May 09 2023

a(18) corrected by Vincenzo Librandi, Nov 05 2011

A038485 Sums of 3 distinct powers of 8.

Original entry on oeis.org

73, 521, 577, 584, 4105, 4161, 4168, 4609, 4616, 4672, 32777, 32833, 32840, 33281, 33288, 33344, 36865, 36872, 36928, 37376, 262153, 262209, 262216, 262657, 262664, 262720, 266241, 266248, 266304, 266752, 294913, 294920, 294976, 295424, 299008, 2097161, 2097217

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base-8 interpretation of A038445.

Cf. A001018, A038484, A038486.

Take[Union[Total/@Subsets[8^Range[0,10],{3}]],40] (* Harvey P. Dale, Jan 31 2016 *)

from math import isqrt, comb
from sympy import integer_nthroot
def A038485(n): return (1<<3*((r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2)))+(1<<3*((a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1)))+(1<<3*(m+t+1)) # Chai Wah Wu, Apr 05 2025

Offset corrected by Amiram Eldar, Jul 14 2022

A038486 Sums of 4 distinct powers of 8.

Original entry on oeis.org

585, 4169, 4617, 4673, 4680, 32841, 33289, 33345, 33352, 36873, 36929, 36936, 37377, 37384, 37440, 262217, 262665, 262721, 262728, 266249, 266305, 266312, 266753, 266760, 266816, 294921, 294977, 294984, 295425, 295432, 295488, 299009, 299016, 299072, 299520, 2097225

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base-8 interpretation of A038446.

Cf. A001018, A038484, A038485.

Sort[Plus @@@ Subsets[8^Range[0, 6], {4}]] (* Amiram Eldar, Jul 14 2022 *)

Offset corrected by Amiram Eldar, Jul 14 2022

A087462 Generalized mod 3 multiplicative Jacobsthal sequence.

Original entry on oeis.org

1, 1, 1, 8, 5, 11, 64, 43, 85, 512, 341, 683, 4096, 2731, 5461, 32768, 21845, 43691, 262144, 174763, 349525, 2097152, 1398101, 2796203, 16777216, 11184811, 22369621, 134217728, 89478485, 178956971, 1073741824, 715827883, 1431655765, 8589934592, 5726623061

Offset: 0

Paul Barry, Sep 08 2003

2^n = a(n) + A087463(n) + A087464(n) provides a decomposition of Pascal's triangle.

Multiplicative analog of A078008.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,8).

Cf. A001045, A001018 (trisection), A082311 (trisection), A082365 (trisection).

Vec(-(4*x^5-2*x^4+x^3+x^2+x+1)/((x+1)*(2*x-1)*(x^2-x+1)*(4*x^2+2*x+1)) + O(x^100)) \\ Colin Barker, Nov 02 2015

a(n) = Sum_{k=0..n} if (mod(n*k, 3)=0, 1, 0) * C(n, k).
a(n) = 2^n-2/3*(1-cos(2*Pi*n/3))*(A001045(n)+2*A001045(n-1)+0^n).
From Colin Barker, Nov 02 2015: (Start)
a(n) = 7*a(n-3)+8*a(n-6) for n>5.
G.f.: -(4*x^5-2*x^4+x^3+x^2+x+1) / ((x+1)*(2*x-1)*(x^2-x+1)*(4*x^2+2*x+1)).
(End)

A135778 Numbers having number of divisors equal to number of digits in base 8.

Original entry on oeis.org

1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 121, 169, 289, 361, 514, 515, 517, 519, 526, 527, 533, 535, 537, 538, 542, 543, 545, 551, 553, 554, 559, 562, 565, 566, 573, 579, 581, 583, 586, 589, 591, 597, 611, 614, 622, 623, 626, 629, 633, 634

Offset: 1

M. F. Hasler, Nov 28 2007

Since 8 is not a prime, no element > 1 of the sequence A001018(k)=8^k (having k+1 digits in base 8, but much more divisors) can be member of this sequence. Also, no power of a prime less than 8 can be in the sequence, since it will always have fewer divisors than digits in base 8. However all powers of 11 up to 11^6 are in this sequence, having the same number of digits (in base 8) than the same power of 8 (since 6 = floor(log(11/8)/log(8))) and also that number of divisors (since 11 is prime).

			a(1) = 1 since 1 has 1 divisor and 1 digit (in base 8 as in any other base).
They are followed by the primes (having 2 divisors {1,p}) between 8 and 8^2 - 1 (to have 2 digits in base 8).
Then come the squares of primes (3 divisors) between 8^2 = 100_8 and 8^3 - 1 = 777_8.
These are followed by all semiprimes and cubes of primes (4 divisors) between 8^3 and 8^4 - 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A135772-A135779, A095862.

Select[Range[1000],IntegerLength[#,8]==DivisorSigma[0,#]&] (* Harvey P. Dale, Mar 04 2016 *)

for(d=1,4,for(n=8^(d-1),8^d-1,d==numdiv(n)&print1(n", ")))

More terms from Harvey P. Dale, Mar 04 2016

A144072 Euler transform of powers of 8.

Original entry on oeis.org

1, 8, 100, 1144, 12906, 141848, 1532276, 16290920, 170938483, 1773107760, 18208004664, 185316171472, 1871103319988, 18756665504080, 186798940872312, 1849265718114736, 18207140415436701, 178355043327697976, 1738966407826985884, 16881111732250394440

Offset: 0

Alois P. Heinz, Sep 09 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

8th column of A144074.

Cf. A001018 (powers of 8).

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:=n-> etr(j->8^j)(n): seq(a(n), n=0..40);

nmax = 20; CoefficientList[Series[Product[1/(1-x^j)^(8^j), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 14 2015 *)

G.f.: Product_{j>0} 1/(1-x^j)^(8^j).
a(n) ~  8^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(8^(m-1)-1)) = 0.0772633520042039151361539536110877247158170... . - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - 8*x^k))). - Ilya Gutkovskiy, Nov 10 2018

A156201 Numerator of Euler(n, 1/8).

Original entry on oeis.org

1, -3, -7, 117, 497, -15123, -95767, 4116837, 34741217, -1921996323, -20273087527, 1370953667157, 17352768515537, -1386843017916723, -20479521468959287, 1888542637550347077, 31872138933891307457, -3331009898404800736323, -63243057486503656319047

Offset: 0

N. J. A. Sloane, Nov 07 2009

T. D. Noe, Table of n, a(n) for n = 0..100

For denominators see A001018. Cf. A000813.

p := proc(n) local j; 2*I*(1+add(binomial(n,j)*polylog(-j,I)*4^j, j=0..n)) end:  A156201 := n -> Im(p(n));
seq(A156201(i), i=0..10);  # Peter Luschny, Apr 29 2013

Table[EulerE[n, 1/8] // Numerator, {n, 0, 18}] (* Jean-François Alcover, Apr 30 2013 *)

a(n) = Im(2*i*(1+Sum_{j=0..n} (binomial(n,j)*Li_{-j}(i)*4^j))). - Peter Luschny, Apr 29 2013
G.f.: conjecture T(0)/(1+3*x), where T(k) = 1 - 16*x^2*(k+1)^2/(16*x^2*(k+1)^2 + (1+3*x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013
a(n) = (-4)^n*skp(n, 3/4), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014
a(n) = 2^(4*n+1)*(zeta(-n,1/16)-zeta(-n, 9/16)), where zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
From Emanuele Munarini, Aug 22 2022: (Start)
E.g.f.: (2*e^t)/(e^(8*t)+1).
E.g.f. for the sequence of the absolute values: (cos(3*t)+sin(3*t))/cos(4*t).
|a(2*n)| = Sum_{k=0..n} binomial(2*n,2*k) (-1)^k 4^(2*n-2*k) 3^(2*k) |E(2*n-2k)|
|a(2*n+1)| = Sum_{k=0..n} binomial(2*n+1,2*k+1) (-1)^k 4^(2*n-2*k) 3^(2*k+1) |E(2*n-2*k)|
where the E(n)'s are the Euler numbers (A122045).
(End)

A156566 a(2n+2) = 9a(2n+1), a(2n+1) = 9a(2n) - 8^n*A000108(n), a(0)=1.

Original entry on oeis.org

1, 8, 72, 640, 5760, 51712, 465408, 4186112, 37675008, 339017728, 3051159552, 27459059712, 247131537408, 2224149233664, 20017343102976, 180155188248576, 1621396694237184, 14592546256715776, 131332916310441984

Offset: 0

Philippe Deléham, Feb 10 2009

nonn

Hankel transform is 8^C(n+1,2).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000108, A001405, A151162, A151254, A151281.
Cf. A156195, A156270, A156361, A156362, A156577.
Cf. A001018, A120730.

a[0] = 1; a[1] = 8; a[2] = 72; a[n_] := a[n] = (-288*(n-2)*a[n-3] + 32*(n-2)*a[n-2] + 9*(n+1)*a[n-1])/(n+1); Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Nov 15 2016 *)
a[n_]:= a[n]= If[n==0, 1, If[OddQ[n], 9*a[n-1] - 8^((n-1)/2)*CatalanNumber[(n- 1)/2], 9*a[n-1]]]; Table[a[n], {n,0,30}] (* G. C. Greubel, May 18 2022 *)

def a(n): # a = A156566
    if (n==0): return 1
    elif (n%2==1): return 9*a(n-1) - 8^((n-1)/2)*catalan_number((n-1)/2)
    else: return 9*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, May 18 2022

a(n) = Sum_{k=0..n} A120730(n,k)*8^k.

A190543 a(n) = 8^n - 3^n.

Original entry on oeis.org

0, 5, 55, 485, 4015, 32525, 261415, 2094965, 16770655, 134198045, 1073682775, 8589757445, 68718945295, 549754219565, 4398041728135, 35184357739925, 281474933663935, 2251799684545085, 18014398122061495, 144115186913594405, 1152921501120062575, 9223372026394422605

Offset: 0

Vincenzo Librandi, Jun 02 2011

Length-n words from letters {1, 2, ..., 8} with at least one letter greater than 3. - Joerg Arndt, Jun 02 2011

All terms are odd multiples of 5, since the powers of 8 mod 10 are 8, 4, 2, 6, ... and the powers of 3 mod 10 are 3, 9, 7, 1, ... - Alonso del Arte, Feb 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
Index entries for linear recurrences with constant coefficients, signature (11,-24).

Cf. A000244, A001018, A016140, A016177.

Magma
```
[8^n - 3^n: n in [0..30]];
```

A190543:=n->8^n - 3^n; seq(A190543(n), n=0..20); # Wesley Ivan Hurt, Feb 26 2014

Table[8^n - 3^n, {n, 0, 19}] (* Alonso del Arte, Feb 25 2014 *)
CoefficientList[Series[5 x/((1 - 3 x) (1 - 8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 05 2014 *)

a(n)=8^n-3^n \\ Charles R Greathouse IV, Jun 02 2011

a(n) = 11*a(n-1) - 24*a(n-2).
a(n) = A001018(n) - A000244(n). - Michel Marcus, Feb 26 2014
From Vincenzo Librandi, Oct 05 2014: (Start)
G.f.: 5*x/((1-3*x)*(1-8*x)).
a(n+1) = 5*A016140(n). (End)
E.g.f.: 2*exp(11*x/2)*sinh(5*x/2). - Elmo R. Oliveira, Mar 31 2025

cp's OEIS Frontend

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

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A199301 a(n) = (2n+1)*8^n.

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A038485 Sums of 3 distinct powers of 8.

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A038486 Sums of 4 distinct powers of 8.

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A087462 Generalized mod 3 multiplicative Jacobsthal sequence.

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A135778 Numbers having number of divisors equal to number of digits in base 8.

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A144072 Euler transform of powers of 8.

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A156201 Numerator of Euler(n, 1/8).

A156566 a(2n+2) = 9a(2n+1), a(2n+1) = 9a(2n) - 8^n*A000108(n), a(0)=1.