A073668 -id:A073668 - cp's OEIS frontend

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

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

A276349 Numbers consisting of a nonempty string of 1's followed by a nonempty string of 0's.

Original entry on oeis.org

10, 100, 110, 1000, 1100, 1110, 10000, 11000, 11100, 11110, 100000, 110000, 111000, 111100, 111110, 1000000, 1100000, 1110000, 1111000, 1111100, 1111110, 10000000, 11000000, 11100000, 11110000, 11111000, 11111100, 11111110, 100000000, 110000000, 111000000

Offset: 1

Jaroslav Krizek, Aug 30 2016

Intersection of A037415 and A009996 except for 1 [Corrected by David A. Corneth, Aug 30 2016].
Set of terms from sequence A052983.
a(n) is the binary expansion of A043569(n). - Michel Marcus, Sep 04 2016

			60 is of the form binomial(a, 2) + b where 0 < b <= a and a = 11, b = 5. So a(60) has (11 + 1) digits and 5 leading ones. The other digits are 0. Giving a(60) = 111110000000. It has 7 (more than 1) trailing zeros so the next one, a(61) is a(60) + 10^(7 - 1). - _David A. Corneth_, Aug 30 2016

Robert Israel, Table of n, a(n) for n = 1..10000
Luboš Pick, Dirichletovy šuplíčky, Pokroky matematiky, fyziky a astronomie, Vol. 61, No. 2 (2016), pp. 106-118. (In Czech; The Dirichlet pigeonhole principle)

Cf. A002024, A009996, A037415, A043569, A052983, A073668, A227362, A276348, A309761.

[n: n in [1..10^7] | Seqint(Setseq(Set(Sort(Intseq(n))))) eq 10 and Seqint(Sort((Intseq(n)))) eq n];

seq(seq(10^(m+1)*(1-10^(-j))/9,j=1..m),m=1..20); # Robert Israel, Sep 02 2016

Table[FromDigits@ Join[ConstantArray[1, #1], ConstantArray[0, #2]] & @@@ Transpose@ {#, n - #} &@ Range[n - 1], {n, 2, 9}] // Flatten (* Michael De Vlieger, Aug 30 2016 *)
Flatten[Table[FromDigits[Join[PadRight[{},n,1],PadRight[{},k,0]]],{n,8},{k,8}]]//Sort (* Harvey P. Dale, Jan 09 2019 *)

is(n) = vecmin(digits(n))==0 && vecmax(digits(n))==1 && digits(n)==vecsort(digits(n), , 4) \\ Felix Fröhlich, Aug 30 2016

a(n) = my(r =  ceil((sqrt(1+8*n)+1)/2), k = n - binomial(r-1, 2));10^(r-k)*(10^(k)-1)/9
\\ given an element n, computes the next element of the sequence.
nxt(n) = my(d = digits(n), qd=#d, s = vecsum(d)); if(qd-s>1, n+10^(qd-s-1), 10^qd)
\\ given an element n of the sequence, computes its place in the sequence.
inv(n) = my(d = digits(n)); binomial(#d-1,2) + vecsum(d) \\ David A. Corneth, Aug 31 2016

from math import isqrt, comb
def A276349(n): return 10*(10**(m:=isqrt(n<<3)+1>>1)-10**(comb(m+1,2)-n))//9 # Chai Wah Wu, Jun 16 2025

A227362(a(n)) = 10.
From Robert Israel, Sep 02 2016: (Start)
a((m^2-m)/2+j) = 10^(m+1)*(1-10^(-j))/9 for m>=1, 1<=j<=m.
a(n) = 10*(10^m - 10^(-n+m*(m+1)/2))/9 where m = A002024(n). (End)
A002275(A002260(n)) * 10^A004736(n) - Peter Kagey, Sep 02 2016
Sum_{n>=1} 1/a(n) = A073668. - Amiram Eldar, Feb 20 2022
a(n) = 10*A309761(n). - Chai Wah Wu, Jun 16 2025

A135702 Decimal expansion of 3*Sum_{k=1..inf} 1/(10^k-1).

Original entry on oeis.org

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

Offset: 0

Eric Desbiaux, Mar 03 2008

Equals: 3*A073668, also A065444/3.

			0.36697273...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A073668, A065444.

RealDigits[9*N[ (1/3) Sum[1/(10^k - 1), {k, 1, Infinity}], 120]] [[1]]

3*suminf(k=1, 1/(10^k-1)) \\ Michel Marcus, Oct 30 2016

A159200 Decimal expansion of Sum_{k >= 1} (1/(10^(4k + 2) - 1)) - (1/(10^(2k + 1) - 1)), negated.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 3, 0, 3, 0, 1, 0, 3, 0, 1, 0, 1, 0, 5, 0, 1, 0, 2, 0, 3, 0, 1, 0, 3, 0, 3, 0, 1, 0, 1, 0, 5, 0, 3, 0, 1, 0, 3, 0, 1, 0, 1, 0, 5, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 3, 0, 1, 0, 3, 0, 3, 0, 3, 0, 1, 0, 5, 0

Offset: 0

Eric Desbiaux, Apr 06 2009

It equals Sum_{k >= 1} 1/((2^(4*k + 2)*5^(4*k + 2)) - 1) - 1/((2^(2*k + 1)*5^(2*k + 1)) - 1).
Note that Sum_{k >= 1} (1/(10^k - 1)) / Sum_{k >= 1} ((1/(10^(4*k + 2) - 1)) -(1/(10^(2*k + 1) - 1))) = A073668 / Sum_{k >= 1} ((1/(10^(4*k + 2) - 1)) - (1/(10^(2*k + 1) - 1))) = -121.100.
My idea for this decimal expansion came from the Engel expansion of e - 1, i.e., A000027(n) = n, and the Engel expansion of e^(-1), i.e., A059193(n) = 2*(2*n + 1)*(n - 1), which I have transformed into (2*n + 1)^2 - (6*n + 3) (since 2*(2*n + 1)*(n - 1) = (2*n + 1)^2 - (6*n + 3)). It appears that the Engel expansion of 1/e works like a Sundaram sieve.
Decimal expansion of Sum_{n>=0} (d(2*n+1) - 1)/(10^(2*n+1) - 1), where d = A000005. - Jianing Song, Apr 12 2021

			-0.00101010201010301010301020301010303010301010501020301030301...

Eric Weisstein's World of Mathematics, Engel expansion.
English Wikipedia, Sieve of Sundaram.
French Wikipedia, Crible de Sundaram.

suminf(k=1, 1/(10^(4*k + 2) - 1) - 1/(10^(2*k + 1) - 1)) \\ Michel Marcus, Jun 25 2019

Comments edited by Petros Hadjicostas, Jun 19 2019

A134687 Successive digits of the numbers A000005(n).

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

Offset: 1

Omar E. Pol, Nov 06 2007

Cf. A000005, A073668, A122667, A122668.

A381788 Greedy expansion of Pi-3 in a base with place values 1/(10^k-1), k >= 1, using digits {0,1,2,...,8,9,A=10}.

Original entry on oeis.org

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

Offset: 1

Simon Plouffe, Mar 07 2025

From Pontus von Brömssen, Mar 13 2025: (Start)
Since the ratio of successive place values is less than 1/10, a digit A=10 is sometimes needed. For example, if 10*A073668-37/33 < x < 1/9, the expansion of x must have an A at the second position after the radix point (for any choice of digits, not only greedy).
The expansion is not unique without specifying greedy choice of digits. For example, the number 11/1000 can be represented both as 0.010898908982... and (non-greedily) as 0.00A989899171... in this system.
For a random number, the probability that the digit A occurs decreases exponentially with the position in the expansion (with greedy choice of digits), so it seems very unlikely that 10 is a term of this sequence.
(End)

Cf. A000796, A073668.

BASEN:= proc(x, b, sgn, k)
local i, j, v, premier, fin, lll, liste, w, baz;
    baz := evalf(b);
    v := abs(frac(evalf(x)));
    fin := trunc(evalf(Digits/log10(b)));
    lll := [seq(i^k*(baz^i + sgn), i = 1 .. fin)];
    liste := [];
    for i to fin do w := trunc(v*lll[i]); v := v - w/lll[i]; liste := [op(liste), w] end do;
    RETURN(liste)
end;
BASEN(Pi-3,10,-1,0);

Sum_{k>=1} a(k)/(10^k - 1) = Pi - 3.

Edited by N. J. A. Sloane, Mar 18 2025

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

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A276349 Numbers consisting of a nonempty string of 1's followed by a nonempty string of 0's.

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A135702 Decimal expansion of 3*Sum_{k=1..inf} 1/(10^k-1).

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A159200 Decimal expansion of Sum_{k >= 1} (1/(10^(4*k + 2) - 1)) - (1/(10^(2*k + 1) - 1)), negated.

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A134687 Successive digits of the numbers A000005(n).

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A381788 Greedy expansion of Pi-3 in a base with place values 1/(10^k-1), k >= 1, using digits {0,1,2,...,8,9,A=10}.

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A159200 Decimal expansion of Sum_{k >= 1} (1/(10^(4k + 2) - 1)) - (1/(10^(2k + 1) - 1)), negated.