A214369 -id:A214369 - cp's OEIS frontend

A024023 a(n) = 3^n - 1.

0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, 59048, 177146, 531440, 1594322, 4782968, 14348906, 43046720, 129140162, 387420488, 1162261466, 3486784400, 10460353202, 31381059608, 94143178826, 282429536480, 847288609442, 2541865828328, 7625597484986, 22876792454960

Offset: 0

N. J. A. Sloane

Number of different directions along lines and hyper-diagonals in an n-dimensional cubic lattice for the attacking queens problem (A036464 in n=2, A068940 in n=3 and A068941 in n=4). The n-dimensional direction vectors have the a(n)+1 Cartesian coordinates (i,j,k,l,...) where i,j,k,l,... = -1, 0, or +1, excluding the zero-vector i=j=k=l=...=0. The corresponding hyper-line count is A003462. - R. J. Mathar, May 01 2006
Total number of sequences of length m=1,...,n with nonzero integer elements satisfying the condition Sum_{k=1..m} |n_k| <= n. See the K. A. Meissner link p. 6 (with a typo: it should be 3^([2a]-1)-1). - Wolfdieter Lang, Jan 21 2008
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x and y are disjoint and either 0) x is a proper subset of y or y is a proper subset of x, or 1) x is not a subset of y and y is not a subset of x. Then a(n) = |R|. - Ross La Haye, Mar 19 2009
Number of neighbors in Moore's neighborhood in n dimensions. - Dmitry Zaitsev, Nov 30 2015
Number of terms in conjunctive normal form of Boolean expression with n variables. E.g., a(2) = 8: [~x, ~y, x, y, ~x|~y, ~x|y, x|~y, x|y]. - Yuchun Ji, May 12 2023
Number of rays of the Coxeter arrangement of type B_n. Equivalently, number of facets of the n-dimensional type B permutahedron. - Jose Bastidas, Sep 12 2023

			From _Zerinvary Lajos_, Jan 14 2007: (Start)
Ternary......decimal:
0...............0
2...............2
22..............8
222............26
2222...........80
22222.........242
222222........728
2222222......2186
22222222.....6560
222222222...19682
2222222222..59048
etc...........etc.
(End)
Sequence combinatorics: n=3: With length m=1: [1],[2],[3] each with 2 signs, with m=2: [1,1], [1,2], [2,1], each 2^2 = 4 times from choosing signs; m=3: [1,1,1] coming in 2^3 signed versions: 3*2 + 3*4 + 1*8 = 26 = a(3). The order is important, hence the M_0 multinomials A048996 enter as factors.
A027902 gives the 384 divisors of a(24). - _Reinhard Zumkeller_, Mar 11 2010

Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Omran Ahmadi and Robert Granger, An efficient deterministic test for Kloosterman sum zeros, Mathematics of Computation, Vol. 83, No. 285 (2014), pp. 347-363; arXiv preprint, arXiv:1104.3882 [math.NT], 2011-2012. See 1st column of Table 2, p. 9.
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.
Michael Baake, Franz Gähler, and Uwe Grimm, Examples of Substitution Systems and Their Factors, Journal of Integer Sequences, Vol. 16 (2013), #13.2.14.
R. Samuel Buss, Herbrand's Theorem, University of California, Logic and Computational Complexity pp. 195-209, Lecture Notes in Computer Science, vol 960. Springer.
Jan Draisma, Tyrrell B. McAllister and Benjamin Nill, Lattice width directions and Minkowski's 3^d-theorem, SIAM J. Discrete Math., Vol. 26, No. 3 (2012), pp. 1104-1107; arXiv preprint, arXiv:0901.1375 [math.CO], Jan 10 2009.
Alessandro Farinelli, Herbrand Universe and Herbrand Base.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Krzysztof A. Meissner, Black hole entropy in Loop Quantum Gravity, Classical and Quantum Gravity, Vol. 21, No. 22 (2004), pp. 5245--5251; arXiv preprint, arXiv:gr-qc/0407052, 2004.
Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20, case three-in-a row, sequence b(n).
Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Wikipedia, Herbrand Structure.
Damiano Zanardini, Computational Logic, Slides, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid, 2009-2010.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. triangle A013609.
Cf. A003462, A007051, A034472, A214369.
Cf. second column of A145901.

a024023 = subtract 1 . a000244  -- Reinhard Zumkeller, Jun 30 2013

[3^n-1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

3^Range[0,30]-1 (* Paolo Xausa, Jul 15 2023 *)

a(n)=3^n-1 \\ Charles R Greathouse IV, Sep 24 2015

vector(50, n, sum(k=0, n, 2^k*binomial(n-1, k))-1) \\ Altug Alkan, Oct 04 2015

my(x='x+O('x^100)); concat([0], Vec(2*x/(-1+x)/(-1+3*x))) \\ Altug Alkan, Oct 16 2015

a(n) = A000244(n) - 1.
a(n) = 2*A003462(n). - R. J. Mathar, May 01 2006
A128760(a(n)) > 0. - Reinhard Zumkeller, Mar 25 2007
G.f.: 2*x/((-1+x)*(-1+3*x)) = 1/(-1+x) - 1/(-1+3*x). - R. J. Mathar, Nov 19 2007
a(n) = Sum_{k=1..n} Sum_{m=1..k} binomial(k-1,m-1)*2^m, n >= 1. a(0)=0. From the sequence combinatorics mentioned above. Twice partial sums of powers of 3.
E.g.f.: e^(3*x) - e^x. - Mohammad K. Azarian, Jan 14 2009
a(n) = A024101(n)/A034472(n). - Reinhard Zumkeller, Feb 14 2009
a(n) = 3*a(n-1) + 2 (with a(0)=0). - Vincenzo Librandi, Nov 19 2010
E.g.f.: -E(0) where E(k) = 1 - 3^k/(1 - x/(x - 3^k*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
a(n) = A227048(n,A020914(n)). - Reinhard Zumkeller, Jun 30 2013
Sum_{n>=1} 1/a(n) = A214369. - Amiram Eldar, Nov 11 2020
a(n) = Sum_{k=1..n} 2^k*binomial(n,k). - Ridouane Oudra, Jun 15 2025
From Peter Bala, Jul 01 2025: (Start)
For n >= 1, a(2*n)/a(n) = A034472(n) and a(3*n)/a(n) = A034513(n).
Modulo differences in offsets, exp( Sum_{n >= 1} a(k*n)/a(n)*x^n/n ) is the o.g.f. of A003462 (k = 2), A006100 (k = 3), A006101 (k = 4), A006102 (k = 5), A022196 (k = 6), A022197 (k = 7), A022198 (k = 8), A022199 (k = 9), A022200 (k = 10), A022201 (k = 11), A022202 (k = 12) and A022203 (k = 13).
The following are all examples of telescoping series:
Sum_{n >= 1} 3^n/(a(n)*a(n+1)) = 1/2^2; Sum_{n >= 1} 3^n/(a(n)*a(n+1)*a(n+2)) = 1/(2*8^2).
In general, for k >= 1, Sum_{n >= 1} 3^n/(a(n)*a(n+1)*...*a(n+k)) = 1/(a(1)*a(2)*...*a(k)*a(k)).
Sum_{n >= 1} 3^n/(a(n)*a(n+2)) = 5/64; Sum_{n >= 1} (-3)^n/(a(n)*a(n+2)) = -3/64.
Sum_{n >= 1} 3^n/(a(n)*a(n+4)) = 703/83200; Sum_{n >= 1} (-3)^n/(a(n)*a(n+4)) = - 417/83200. (End)

A073668 Decimal expansion of Sum_{k>=1} 1/(10^k - 1).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 3, 0, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 3, 2, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 3, 2, 2, 4, 6, 6, 4, 8, 3, 0, 5, 4, 3, 2, 4, 4, 4, 8, 3, 2, 4, 6, 4, 4, 5, 2, 2, 6, 6, 9, 2, 8, 2, 8, 8

Offset: 0

Robert G. Wilson v, Aug 29 2002

Parallels A000005 up to a(46).

Sum_{k>=1} x^k/(1-x^k) = Sum_{k>=1} tau(k)*x^k. Choosing x = 1/10 gives the result. - Amarnath Murthy, Oct 21 2002

			0.122324243426244526264428344628264449244... = A065444/9.

Amarnath Murthy, Some interesting results on d(N), the number of divisors of a natural number, page 463, Octogon Mathematical Magazine, Vol. 8 No. 2, October 2000.

Cf. A065442, A214369, A248721, A248722, A248723, A248724, A248725, A248726.

evalf(Sum(1/(10^k - 1), k = 1..infinity), 200) # Vaclav Kotesovec, Jul 16 2019
# second program with faster converging series after Joerg Arndt
evalf( add( (1/10)^(n^2)*(1 + 2/(10^n - 1)), n = 1..8), 105); # Peter Bala, Jan 30 2022

RealDigits[ N[ Sum[1/(10^k - 1), {k, 1, Infinity}], 120]] [[1]]

suminf(k=1,1/(10^k-1)) \\ Charles R Greathouse IV, Oct 05 2014

From Eric Desbiaux, Mar 11 2009: (Start)
Equals Sum_{k >= 1} 1/((2^k*5^k)-1).
Equals Sum_{k >= 1} (1/2^k)*(1/5^k)/(1-((1/2^k)*(1/5^k))).
Sum_{k >= 1} 1/(5^k) = 1/4.
Sum_{k >= 1} 1/(2^k) = 1.
Sum_{k >= 1} (1/5^k)/(1-((1/2^k)*(1/5^k))) = 0.2726344339156...
Sum_{k >= 1} (1/2^k)/(1-((1/2^k)*(1/5^k))) = 1.0582125127815...
Sum_{k >= 1} 1/(1-((1/2^k)*(1/5^k))) - 1 = A073668.
(End)
Fast computation via Lambert series: 0.122324243426... = Sum_{n>=1} x^(n^2)*(1+x^n)/(1-x^n) where x=1/10. - Joerg Arndt, Oct 18 2020

A248721 Decimal expansion of Sum_{k>=1} 1/(4^k - 1).

Original entry on oeis.org

4, 2, 1, 0, 9, 7, 6, 8, 6, 0, 3, 3, 4, 2, 3, 7, 7, 7, 2, 9, 5, 9, 9, 0, 8, 8, 7, 9, 6, 7, 7, 1, 3, 0, 4, 8, 9, 6, 1, 4, 4, 1, 3, 3, 6, 3, 2, 4, 1, 1, 5, 4, 0, 4, 6, 0, 5, 9, 2, 0, 7, 9, 6, 7, 1, 2, 7, 7, 1, 3, 7, 0, 4, 8, 8, 7, 3, 9, 8, 0, 2, 7, 5, 1, 9, 0, 3, 6, 8, 4, 7, 5, 8, 6, 5, 0, 7, 9, 5, 3, 9, 2, 8, 4, 5

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.4210976860334237772959908879677130489614413363241154046059207967127713704887...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000005, A065442, A073668, A214369, A248722, A248723, A248724, A248725, A248726.

evalf(sum(1/(4^k-1), k=1..infinity),120) # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series after Joerg Arndt
evalf( add( (1/4)^(n^2)*(1 + 2/(4^n - 1)), n = 1..13), 105); # Peter Bala, Jan 30 2022

x = 1/4; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of Amarnath Murthy, see A073668 *)

suminf(k=1, 1/(4^k-1)) \\ Michel Marcus, Oct 18 2014

Equals Sum_{k>=1} x^(k^2)*(1+x^k)/(1-x^k) where x = 1/4 (the Lambert series evaluated at 1/4). - Joerg Arndt, Jun 03 2020

Equals Sum_{k>=1} d(k)/4^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A048328 Numbers that are repdigits in base 3.

Original entry on oeis.org

0, 1, 2, 4, 8, 13, 26, 40, 80, 121, 242, 364, 728, 1093, 2186, 3280, 6560, 9841, 19682, 29524, 59048, 88573, 177146, 265720, 531440, 797161, 1594322, 2391484, 4782968, 7174453, 14348906, 21523360, 43046720, 64570081, 129140162, 193710244, 387420488, 581130733

Offset: 0

Patrick De Geest, Feb 15 1999

Case for base 2 see A000225: 2^n - 1.

If the sequence b(n) represents the number of paths of length n, n >= 1, starting at node 1 and ending at nodes 1, 2, 3 and 4 on the path graph P_5 then a(n-1) = b(n) - 1. - Johannes W. Meijer, May 29 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Repdigit.
Index entries for 3-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).

Bisections: A024023 and A003462.

Cf. A000225, A010785, A028987, A028988, A033016, A038754, A068911, A124302, A214369.

nmax := 35; a(0) := 0: for n from 1 to nmax do a(2*n) := a(2*n-2) + 2*3^(n-1); od: a(1) := 1: for n from 1 to nmax do a(2*n+1) := 1*a(2*n-1) + 3^n; od: seq(a(n), n=0..nmax);
# End program 1
with(GraphTheory): G := PathGraph(5): A:= AdjacencyMatrix(G): nmax := nmax; for n from 1 to nmax+1 do B(n) := A^n; b(n) := add(B(n)[1, k], k=1..4); a1(n-1) := b(n)-1; od: seq(a1(n), n=0..nmax);
# End program 2
# From Johannes W. Meijer, May 29 2010, revised Sep 23 2012
# third Maple program:
a:= n->(<<0|1>, <-3|4>>^iquo(n, 2, 'r').`if`(r=0, <<0, 2>>, <<1, 4>>))[1, 1]:
seq (a(n), n=0..60);  # Alois P. Heinz, Sep 23 2012

Rest[FromDigits[#, 3]&/@Flatten[Table[{PadRight[{1}, n, 1], PadRight[{2}, n, 2]}, {n, 0, 20}], 1]] (* Harvey P. Dale, Feb 03 2011 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -3,0,4,0]^n*[0;1;2;4])[1,1] \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (2*x^2+x)/(1-4*x^2+3*x^4). - Alois P. Heinz, Sep 23 2012
Sum_{n>=1} 1/a(n) = 3 * A214369 = 2.04646050781571420028... - Amiram Eldar, Jan 21 2022
a(n) = (3^(n/2)*(sqrt(3) + 2 - (-1)^n*(sqrt(3) - 2)) - 3 - (-1)^n)/4. - Stefano Spezia, Feb 18 2022

A248722 Decimal expansion of Sum_{k>=1} 1/(5^k - 1).

Original entry on oeis.org

3, 0, 1, 7, 3, 3, 8, 5, 3, 5, 9, 7, 9, 7, 2, 4, 5, 7, 9, 4, 8, 1, 6, 2, 1, 5, 9, 3, 9, 3, 9, 9, 1, 1, 9, 2, 6, 2, 3, 0, 0, 9, 4, 3, 1, 5, 1, 7, 1, 5, 7, 7, 2, 0, 3, 9, 5, 7, 9, 1, 9, 2, 3, 3, 1, 8, 3, 7, 9, 8, 2, 5, 8, 9, 2, 0, 3, 4, 3, 3, 5, 2, 7, 5, 8, 5, 9, 4, 9, 2, 9, 7, 8, 7, 5, 8, 1, 6, 9, 6, 8, 3, 5, 5, 7

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.301733853597972457948162159393991192623009431517157720395791923318379825892...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000005, A065442, A073668, A214369, A248721, A248723, A248724, A248725, A248726.

evalf( add( (1/5)^(n^2)*(1 + 2/(5^n - 1)), n = 1..12), 105); # Peter Bala, Jan 30 2022

x = 1/5; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of Amarnath Murthy, see A073668 *)

sumpos(k=1,1/(5^k-1)) \\ M. F. Hasler, Oct 15 2014

Equals Sum_{k>=1} d(k)/5^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A248723 Decimal expansion of the Sum_{k>=1} 1/(6^k - 1).

Original entry on oeis.org

2, 3, 4, 1, 4, 9, 1, 3, 0, 1, 3, 4, 8, 0, 9, 2, 0, 6, 4, 8, 5, 1, 1, 1, 6, 7, 2, 8, 1, 3, 8, 7, 2, 9, 1, 8, 5, 4, 6, 3, 6, 1, 0, 3, 4, 7, 8, 6, 5, 1, 3, 8, 9, 8, 5, 2, 2, 4, 2, 1, 3, 8, 6, 7, 1, 0, 2, 3, 8, 1, 9, 8, 6, 6, 2, 8, 7, 9, 2, 3, 2, 2, 5, 6, 7, 8, 8, 7, 9, 5, 0, 1, 8, 7, 8, 3, 9, 1, 2, 6, 6, 5, 5, 3, 4

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.2341491301348092064851116728138729185463610347865138985224213867102381986628...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248724, A248725, A248726.

evalf(sum(1/(6^k-1), k=1..infinity),120); # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/6)^(n^2)*(1 + 2/(6^n - 1)), n = 1..11), 105); # Peter Bala, Jan 30 2022

x = 1/6; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of Amarnath Murthy, see A073668 *)

suminf(k=1, 1/(6^k-1)) \\ Michel Marcus, Oct 18 2014

Equals Sum_{k>=1} d(k)/6^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A248724 Decimal expansion of Sum_{k>=1} 1/(7^k - 1).

Original entry on oeis.org

1, 9, 0, 9, 1, 0, 0, 6, 2, 4, 1, 0, 2, 6, 1, 5, 7, 8, 2, 0, 2, 1, 9, 9, 6, 4, 4, 4, 1, 7, 6, 9, 1, 1, 6, 8, 7, 6, 9, 2, 6, 8, 4, 7, 6, 0, 0, 8, 2, 6, 6, 4, 0, 8, 3, 3, 4, 7, 7, 1, 1, 0, 8, 6, 4, 0, 9, 9, 9, 6, 7, 5, 5, 8, 4, 6, 3, 0, 1, 4, 4, 0, 3, 8, 0, 0, 9, 1, 1, 6, 1, 6, 5, 9, 7, 0, 9, 1, 1, 9, 3, 4, 5, 6, 1

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.1909100624102615782021996444176911687692684760082664083347711086409996755846...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248723, A248725, A248726.

evalf(sum(1/(7^k-1), k=1..infinity),120) # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/7)^(n^2)*(1 + 2/(7^n - 1)), n = 1..11), 105); # Peter Bala, Jan 30 2022

x = 1/7; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of Amarnath Murthy, see A073668 *)

suminf(k=1, 1/(7^k-1)) \\ Michel Marcus, Oct 18 2014

Equals Sum_{k>=1} d(k)/7^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A248725 Decimal expansion of Sum_{k>=1} 1/(8^k - 1).

Original entry on oeis.org

1, 6, 0, 9, 6, 6, 1, 8, 4, 3, 1, 5, 0, 6, 2, 3, 9, 6, 8, 0, 5, 3, 0, 2, 5, 6, 4, 1, 4, 3, 6, 4, 2, 8, 8, 5, 5, 5, 0, 7, 4, 3, 8, 5, 6, 0, 2, 5, 3, 2, 8, 3, 4, 6, 3, 6, 0, 8, 3, 5, 9, 1, 8, 6, 4, 7, 8, 2, 3, 9, 4, 0, 8, 5, 8, 0, 0, 6, 3, 6, 9, 1, 7, 7, 9, 2, 3, 4, 5, 3, 1, 0, 0, 9, 3, 2, 5, 4, 0, 2, 5, 2, 9, 6, 4

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.16096618431506239680530256414364288555074385602532834636083591864782394085800...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248723, A248724, A248726.

evalf(sum(1/(8^k-1), k=1..infinity),120) # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/8)^(n^2)*(1 + 2/(8^n - 1)), n = 1..10), 105); # Peter Bala, Jan 30 2022

x = 1/8; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of Amarnath Murthy, see A073668 *)

suminf(k=1, 1/(8^k-1)) \\ Michel Marcus, Oct 18 2014

Equals Sum_{k>=1} d(k)/8^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A248726 Decimal expansion of Sum_{k>=1} 1/(9^k - 1).

Original entry on oeis.org

1, 3, 9, 0, 4, 5, 1, 1, 7, 6, 6, 2, 1, 8, 8, 1, 2, 9, 3, 5, 8, 7, 2, 8, 4, 7, 4, 3, 6, 9, 0, 8, 9, 0, 5, 2, 1, 3, 9, 3, 6, 2, 6, 4, 7, 0, 6, 7, 8, 1, 9, 6, 0, 9, 5, 5, 1, 0, 3, 5, 4, 9, 3, 4, 7, 9, 6, 7, 0, 2, 0, 1, 4, 5, 3, 6, 6, 6, 6, 0, 7, 9, 8, 8, 6, 3, 3, 7, 9, 8, 1, 3, 5, 7, 6, 5, 5, 0, 5, 7, 9, 9, 5, 5, 3

Offset: 0

Robert G. Wilson v, Oct 12 2014

			0.13904511766218812935872847436908905213936264706781960955103549347967020145366...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248723, A248724, A248725.

evalf(sum(1/(9^k-1), k=1..infinity),120) # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/9)^(n^2)*(1 + 2/(9^n - 1)), n = 1..10), 105); # Peter Bala, Jan 30 2022

x = 1/9; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of Amarnath Murthy, see A073668 *)

suminf(k=1, 1/(9^k-1)) \\ Michel Marcus, Oct 18 2014

Equals Sum_{k>=1} d(k)/9^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 22 2020

A088838 Numerator of the quotient sigma(3n)/sigma(n).

Original entry on oeis.org

4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 121, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 121, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 40, 4, 4, 13, 4, 4, 13, 4, 4, 364, 4, 4, 13, 4, 4, 13, 4, 4, 40

Offset: 1

Labos Elemer, Nov 04 2003

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A007949, A038712, A088837, A088839, A088840, A080278 (denominator), A144613.

Cf. A001620, A214369.

A088838 := proc(n)
    numtheory[sigma](3*n)/numtheory[sigma](n) ;
    numer(%) ;
end proc:
seq(A088838(n),n=1..100) ; # R. J. Mathar, Nov 19 2017
seq((3^(2+padic:-ordp(n,3))-1)/2, n=1..100); # Robert Israel, Nov 19 2017

k=3; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]

a(n) = numerator(sigma(3*n)/sigma(n)) \\ Felix Fröhlich, Nov 19 2017

From Robert Israel, Nov 19 2017: (Start)
a(n) = (3^(2+A007949(n))-1)/2.
G.f.: Sum_{k>=0} (3^(k+2)-1)*(x^(3^k)+x^(2*3^k))/(2*(1-x^(3^(k+1)))). (End)
a(n) = sigma(3*n)/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
From Amiram Eldar, Jan 06 2023: (Start)
a(n) = numerator(A144613(n)/A000203(n)).
Sum_{k=1..n} a(k) ~ (3/log(3))*n*log(n) + (1/2 + 3*(gamma-1)/log(3))*n, where gamma is Euler's constant (A001620).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A080278(k) = 4*A214369 + 1 = 3.728614... . (End)

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