A073668 -id:A073668 - cp's OEIS frontend

A001221 Number of distinct primes dividing n (also called omega(n)).

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 3, 2

Offset: 1

N. J. A. Sloane

From Peter C. Heinig (algorithms(AT)gmx.de), Mar 08 2008: (Start)
This is also the number of maximal ideals of the ring (Z/nZ,+,*). Since every finite integral domain must be a field, every prime ideal of Z/nZ is a maximal ideal and since in general each maximal ideal is prime, there are just as many prime ideals as maximal ones in Z/nZ, so the sequence gives the number of prime ideals of Z/nZ as well.
The reason why this number is given by the sequence is that the ideals of Z/nZ are precisely the subgroups of (Z/nZ,+). Hence for an ideal to be maximal it has form a maximal subgroup of (Z/nZ,+) and this is equivalent to having prime index in (Z/nZ) and this is equivalent to being generated by a single prime divisor of n.
Finally, all the groups arising in this way have different orders, hence are different, so the number of maximal ideals equals the number of distinct primes dividing n. (End)
Equals double inverse Mobius transform of A143519, where A051731 = the inverse Mobius transform. - Gary W. Adamson, Aug 22 2008
a(n) is the number of unitary prime power divisors of n (not including 1). - Jaroslav Krizek, May 04 2009 [corrected by Ilya Gutkovskiy, Oct 09 2019]
Sum_{d|n} 2^(-A001221(d) - A001222(n/d)) = Sum_{d|n} 2^(-A001222(d) - A001221(n/d)) = 1 (see Dressler and van de Lune link). - Michel Marcus, Dec 18 2012
Up to 2*3*5*7*11*13*17*19*23*29  - 1 = 6469693230 - 1, also the decimal expansion of the constant 0.01111211... = Sum_{k>=0} 1/(10 ^ A000040(k) - 1) (see A073668). - Eric Desbiaux, Jan 20 2014
The average order of a(n): Sum_{k=1..n} a(k) ~ Sum_{k=1..n} log log k. - Daniel Forgues, Aug 13-16 2015
From Peter Luschny, Jul 13 2023: (Start)
We can use A001221 and A001222 to classify the positive integers as follows.
   A001222(n) = A001221(n) = 0 singles out {1}.
Restricting to n > 1:
   A001222(n)^A001221(n) = 1: A000040, prime numbers.
   A001221(n)^A001222(n) = 1: A246655, prime powers.
   A001222(n)^A001221(n) > 1: A002808, the composite numbers.
   A001221(n)^A001222(n) > 1: A024619, complement of A246655.
   n^(A001222(n) - A001221(n)) = 1: A144338, products of distinct primes. (End)
Inverse Möbius transform of the characteristic function of primes (A010051). - Wesley Ivan Hurt, Jun 22 2024
Dirichlet convolution of A010051(n) and 1. - Wesley Ivan Hurt, Jul 15 2025

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 48-57.
J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/Cambridge University Press London 1935.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 from NJAS)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Henry Bottomley, Prime factors calculator
J. Brennen, Prime Factoring Applet
J. Britton, Prime Factorization Machine
A. Dendane, Prime Factors Calculator
Robert E. Dressler and Jan van de Lune, Some remarks concerning the number theoretic functions omega and Omega, Proc. Amer. Math. Soc. 41 (1973), 403-406
J. Flament, Decomposition d'un nombre en facteurs premiers
G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92. Also Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI (2000): 262-275.
A. Hodges, Java Applet for Factorisation
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
S. O. S. Math, Prime factorization of the first 1000 integers
K. Matthews, Factorization and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)
J. Moyer, "Prime Factors of Integers" server for numbers up to 10^36
Primefan, The First 2500 Integers,Factored
Primefan, Factorer
F. Richman, Factoring into Primes
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Moebius Transform
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Wikipedia, Prime factor
Wikipedia, Table of prime factors
G. Xiao, WIMS server, Factoris
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

Cf. A001222 (primes counted with multiplicity), A046660, A285577, A346617. Partial sums give A013939.
Cf. also A125666, A069010, A087624, A091202, A091221, A143519, A144494, A158210, A156542, A156552, A000010, A008683.
Sum of the k-th powers of the primes dividing n for k=0..10: this sequence (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), A351198 (k=10).
Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k=0..10: this sequence (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

import Math.NumberTheory.Primes.Factorisation (factorise)
a001221 = length . snd . unzip . factorise
-- Reinhard Zumkeller, Nov 28 2015

using Nemo
function NumberOfPrimeFactors(n; distinct=true)
    distinct && return length(factor(ZZ(n)))
    sum(e for (p, e) in factor(ZZ(n)); init=0)
end
println([NumberOfPrimeFactors(n) for n in 1:60]) # Peter Luschny, Jan 02 2024

[#PrimeDivisors(n): n in [1..120]]; // Bruno Berselli, Oct 15 2021

A001221 := proc(n) local t1, i; if n = 1 then return 0 else t1 := 0; for i to n do if n mod ithprime(i) = 0 then t1 := t1 + 1 end if end do end if; t1 end proc;
A001221 := proc(n) nops(numtheory[factorset](n)) end proc: # Emeric Deutsch
omega := n -> NumberTheory:-NumberOfPrimeFactors(n, 'distinct'): # Peter Luschny, Jun 15 2025

Array[ Length[ FactorInteger[ # ] ]&, 100 ]
PrimeNu[Range[120]]  (* Harvey P. Dale, Apr 26 2011 *)

func(nops(numlib::primedivisors(n)), n):

numlib::omega(n)$ n=1..110 // Zerinvary Lajos, May 13 2008

PARI
```
a(n)=omega(n)
```

from sympy.ntheory import primefactors
print([len(primefactors(n)) for n in range(1, 1001)])  # Indranil Ghosh, Mar 19 2017

def A001221(n): return sum(1 for p in divisors(n) if is_prime(p))
[A001221(n) for n in (1..80)] # Peter Luschny, Feb 01 2012

[sloane.A001221(n) for n in (1..111)] # Giuseppe Coppoletta, Jan 19 2015

[gp.omega(n) for n in range(1,101)] # G. C. Greubel, Jul 13 2024

G.f.: Sum_{k>=1} x^prime(k)/(1-x^prime(k)). - Benoit Cloitre, Apr 21 2003; corrected by Franklin T. Adams-Watters, Sep 01 2009
Dirichlet generating function: zeta(s)*primezeta(s). - Franklin T. Adams-Watters, Sep 11 2005
Additive with a(p^e) = 1.
a(1) = 0, a(p) = 1, a(pq) = 2, a(pq...z) = k, a(p^k) = 1, where p, q, ..., z are k distinct primes and k natural numbers. - Jaroslav Krizek, May 04 2009
a(n) = log_2(Sum_{d|n} mu(d)^2). - Enrique Pérez Herrero, Jul 09 2012
a(A002110(n)) = n, i.e., a(prime(n)#) = n. - Jean-Marc Rebert, Jul 23 2015
a(n) = A091221(A091202(n)) = A069010(A156552(n)). - Antti Karttunen, circa 2004 & Mar 06 2017
L.g.f.: -log(Product_{k>=1} (1 - x^prime(k))^(1/prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = log_2(Sum_{k=1..n} mu(gcd(n,k))^2/phi(n/gcd(n,k))) = log_2(Sum_{k=1..n} mu(n/gcd(n,k))^2/phi(n/gcd(n,k))), where phi = A000010 and mu = A008683. - Richard L. Ollerton, May 13 2021
Sum_{k=1..n} 2^(-a(gcd(n,k)) - A001222(n/gcd(n,k)))/phi(n/gcd(n,k)) = Sum_{k=1..n} 2^(-A001222(gcd(n,k)) - a(n/gcd(n,k)))/phi(n/gcd(n,k)) = 1, where phi = A000010. - Richard L. Ollerton, May 13 2021
a(n) = A005089(n) + A005091(n) + A059841(n) = A005088(n) +A005090(n) +A079978(n). - R. J. Mathar, Jul 22 2021
From Wesley Ivan Hurt, Jun 22 2024: (Start)
a(n) = Sum_{p|n, p prime} 1.
a(n) = Sum_{d|n} c(d), where c = A010051. (End)

A002283 a(n) = 10^n - 1.

Original entry on oeis.org

0, 9, 99, 999, 9999, 99999, 999999, 9999999, 99999999, 999999999, 9999999999, 99999999999, 999999999999, 9999999999999, 99999999999999, 999999999999999, 9999999999999999, 99999999999999999, 999999999999999999, 9999999999999999999, 99999999999999999999, 999999999999999999999, 9999999999999999999999

Offset: 0

N. J. A. Sloane

A friend from Germany remarks that the sequence 9, 99, 999, 9999, 99999, 999999, ... might be called the grumpy German sequence: nein!, nein! nein!, nein! nein! nein!, ...
The Regan link shows that integers of the form 10^n -1 have binary representations with exactly n trailing 1 bits. Also those integers have quinary expressions with exactly n trailing 4's. For example, 10^4 -1 = (304444)5. The first digits in quinary correspond to the number 2^n -1, in our example (30)5 = 2^4 -1. A similar pattern occurs in the binary case. Consider 9 = (1001)2. - Washington Bomfim Dec 23 2010
a(n) is the number of positive integers with less than n+1 digits. - Bui Quang Tuan, Mar 09 2015
From Peter Bala, Sep 27 2015: (Start)
For n >= 1, the simple continued fraction expansion of sqrt(a(2*n)) = [10^n - 1; 1, 2*(10^n - 1), 1, 2*(10^n - 1), ...] has period 2. The simple continued fraction expansion of sqrt(a(2*n))/a(n) = [1; 10^n - 1, 2,  10^n - 1, 2, ...] also has period 2. Note the occurrence of large partial quotients in both expansions.
A theorem of Kuzmin in the measure theory of continued fractions says that large partial quotients are the exception in continued fraction expansions.
Empirically, we also see the presence of unexpectedly large partial quotients early in the continued fraction expansions of the m-th roots of the numbers a(m*n) for m >= 3. Some typical examples are given below. (End)
For n > 0, numbers whose smallest decimal digit is 9. - Stefano Spezia, Nov 16 2023

			From _Peter Bala_, Sep 27 2015: (Start)
Continued fraction expansions showing large partial quotients:
a(12)^(1/3) = [9999; 1, 299999998, 1, 9998, 1, 449999998, 1, 7998, 1, 535714284, 1, 2, 2, 142, 2, 2, 1, 599999999, 3, 1, 1,...].
Compare with a(30)^(1/3) = [9999999999; 1, 299999999999999999998, 1, 9999999998, 1, 449999999999999999998, 1, 7999999998, 1, 535714285714285714284, 1, 2, 2, 142857142, 2, 2, 1, 599999999999999999999, 3, 1, 1,...].
a(24)^(1/4) = [999999; 1, 3999999999999999998, 1, 666665, 1, 1, 1, 799999999999999999, 3, 476190, 7, 190476190476190476, 21, 43289, 1, 229, 1, 1864801864801863, 1, 4, 6,...].
Compare with a(48)^(1/4) = [999999999999; 1, 3999999999999999999999999999999999998, 1, 666666666665, 1, 1, 1, 799999999999999999999999999999999999, 3, 476190476190, 7, 190476190476190476190476190476190476, 21, 43290043289, 1, 229, 1, 1864801864801864801864801864801863, 1, 4, 6,...].
a(25)^(1/5) = [99999, 1, 499999999999999999998, 1, 49998, 1, 999999999999999999998, 1, 33332, 3, 151515151515151515151, 5, 1, 1, 1947, 1, 1, 38, 3787878787878787878, 1, 3, 5,...].
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Peter Bala, A002283 and some continued fraction expansions.
Rick Regan, Nines in quinary, September 8th, 2009.
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.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A000533, A003020, A007138, A007953, A008591, A010888, A048379, A066138, A073668, A168624, A276352.
Cf. A002275, A002276, A002277, A002278, A002279, A002280, A002281, A002282.
Cf. A075412, A075415, A178630, A178631, A178632, A178633, A178634, A178635.
Cf. A010785.

a002283 = subtract 1 . (10 ^)  -- Reinhard Zumkeller, Feb 21 2014

[(10^n-1): n in [0..20]]; // Vincenzo Librandi, Apr 26 2011

Table[10^n - 1, {n, 0, 22}] (* Michael De Vlieger, Sep 27 2015 *)

A002283(n):=10^n-1$
makelist(A002283(n),n,0,20); /* Martin Ettl, Nov 08 2012 */

a(n)=10^n-1; \\ Charles R Greathouse IV, Jan 30 2012

def a(n): return 10**n-1 # Michael S. Branicky, Feb 25 2023

From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-10*x)-1/(1-x).
E.g.f.: e^(10*x)-e^x. (End)
a(n) = A075412(n)/A002275(n) = A178630(n)/A002276(n) = A178631(n)/A002277(n) = A075415(n)/A002278(n) = A178632(n)/A002279(n) = A178633(n)/A002280(n) = A178634(n)/A002281(n) = A178635(n)/A002282(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 9*10^(n-1) with a(0)=0; Also: a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0, a(1)=9. - Vincenzo Librandi, Jul 22 2010
For n>0, A007953(a(n)) = A008591(n) and A010888(a(n)) = 9. - Reinhard Zumkeller, Aug 06 2010
A048379(a(n)) = 0. - Reinhard Zumkeller, Feb 21 2014
a(n) = Sum_{k=1..n} 9*10^k. - Carauleanu Marc, Sep 03 2016
Sum_{n>=1} 1/a(n) = A073668. - Amiram Eldar, Nov 13 2020
From Elmo R. Oliveira, Jul 19 2025: (Start)
a(n) = 9*A002275(n).
a(n) = A010785(A008591(n)). (End)

More terms from Michael De Vlieger, Sep 27 2015

A065442 Decimal expansion of Erdős-Borwein constant Sum_{k>=1} 1/(2^k - 1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 18 2001

Also the decimal expansion of the (finite) value of Sum_{ k >= 1, k has no digit equal to 0 in base 2 } 1/k. - Robert G. Wilson v, Aug 03 2010

This constant is irrational (Erdős, 1948; Borwein, 1992). - Amiram Eldar, Aug 01 2020

			1.60669515241529176378330152319092458048057967150575643577807955369...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, p. 258.

Harry J. Smith, Table of n, a(n) for n = 1..2000
David H. Bailey and Richard E. Crandall, Random generators and normal numbers, Experimental Mathematics, Vol. 11, No. 4 (2002), pp. 527-546.
Robert Baillie, Summing The Curious Series Of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Peter Borwein, On the Irrationality of Certain Series, Math. Proc. Cambridge Philos. Soc., Vol. 112, No. 1 (1992), pp. 141-146, alternative link.
Richard Crandall, The googol-th bit of the Erdős-Borwein constant, Integers, 12 (2012), A23.
Paul Erdős, On Arithmetical Properties of Lambert Series, J. Indian Math. Soc., Vol. 12 (1948), 63-66.
Steven R. Finch, Digital Search Tree Constants [Broken link]
Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]
Nobushige Kurokawa and Yuichiro Taguchi, A p-analogue of Euler’s constant and congruence zeta functions, Proc. Japan Acad. Ser. A Math. Sci., Volume 94, Number 2 (2018), 13-16.
Mathematics Stack Exchange, Find Sum_{k = 1..oo} 1/(2^(k+1) - 1).
Yohei Tachiya, Irrationality of Certain Lambert Series, Tokyo J. Math. 27 (1) 75 - 85, June 2004.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, arXiv preprint arXiv:1608.00795 [math.NT], 2016.
Eric Weisstein's Mathworld, Erdos-Borwein Constant, Tree Searching, Double Series, Irrational Number
Hengjie Yang and Richard D. Wesel, Systematic Transmission With Fountain Parity Checks for Erasure Channels With Stop Feedback, arXiv:2307.14507 [cs.IT], 2023.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, J. Int. Seq., Vol. 23 (2020), Article 20.10.4.

See A038631 for continued fraction.

Cf. A000005, A000203 (see formulas), A065443, A066766, A179951, A211705, A211706, A323482.

# Uses Lambert series, cf. formula by Arndt:
evalf( add( (1/2)^(n^2)*(1 + 2/(2^n - 1)), n = 1..20 ), 105);
# Peter Bala, Jan 22 2021

RealDigits[ Sum[1/(2^k - 1), {k, 350}], 10, 111][[1]] (* Robert G. Wilson v, Nov 05 2006 *)
(* first install irwinSums.m, see reference, then *) First@ RealDigits@ iSum[0, 0, 111, 2] (* Robert G. Wilson v, Aug 03 2010 *)
RealDigits[(Log[2] - 2 QPolyGamma[0, 1, 2])/Log[4], 10, 100][[1]] (* Fred Daniel Kline, May 23 2011 *)
x = 1/2; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* Robert G. Wilson v, Oct 12 2014 after an observation and formula of Amarnath Murthy, see A073668 *)

a(n)= s=0; for(x=1,n,s=s+1.0/(2^x-1)); s

default(realprecision, 2080); x=suminf(k=1, 1/(2^k - 1)); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065442.txt", n, " ", d)) \\ Harry J. Smith, Oct 19 2009

k=1.; suminf(n=1, k>>=1; k^n*(1+k)/(1-k)) \\ Charles R Greathouse IV, Jun 03 2015

Note: Sum_{k>=1} d(k)/2^k = Sum_{k>=1} 1/(2^k - 1).
Fast computation via Lambert series: 1.60669515... = Sum_{n>=1} x^(n^2)*(1+x^n)/(1-x^n) where x=1/2. - Joerg Arndt, May 24 2011
Equals (1/2) * A211705. - Amiram Eldar, Aug 01 2020
Equals 1/4 + Sum_{k >= 2} (1 + 8^k)/((2^k - 1)*2^(k^2+k)). See Mathematics Stack Exchange link. - Peter Bala, Jan 28 2022
Equals A066766 - A065443. - Amiram Eldar, Oct 16 2022

More terms from Randall L Rathbun, Jan 16 2002

A214369 Decimal expansion of Sum_{n>=1} 1/(3^n-1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 14 2012

			Equals 0.6821535026052380667...

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

Cf. A024023, A065442, A073668, A248721, A248722, A248723, A248724, A248725, A248726.

evalf(sum(1/(3^k-1), k=1..infinity), 120); # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/3)^(n^2)*(1 + 2/(3^n - 1)), n = 1..14 ), 105); # Peter Bala, Jan 30 2022

RealDigits[ NSum[1/(3^n - 1), {n, 1, Infinity}, WorkingPrecision -> 110, NSumTerms -> 100], 10, 105] // First (* or *) 1 - (Log[2] + QPolyGamma[0, 1, 1/3])/Log[3] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Jun 05 2013 *)
x = 1/3; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* Robert G. Wilson v, Oct 12 2014 after an observation and the formula of Amarnath Murthy, see A073668 *)

suminf(n=1, 1/(3^n-1)) \\ Michel Marcus, Mar 11 2017

Equals Sum_{n>=1} 1/A024023(n).

Equals Sum_{k>=1} d(k)/3^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, May 17 2020

More terms from Jean-François Alcover, Feb 12 2013

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

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

cp's OEIS Frontend

A001221 Number of distinct primes dividing n (also called omega(n)).

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A002283 a(n) = 10^n - 1.

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A065442 Decimal expansion of Erdős-Borwein constant Sum_{k>=1} 1/(2^k - 1).

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A214369 Decimal expansion of Sum_{n>=1} 1/(3^n-1).

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

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