A052268 -id:A052268 - cp's OEIS frontend

A171386 The characteristic function of 2 and 3: 1 if n is prime such that either n-1 or n+1 is prime, else 0.

0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Juri-Stepan Gerasimov, Dec 07 2009

A181354(n) + A181376(n) + A181378(n) + A181380(n) + A181384(n) + A181401(n) + A181403(n) + A181405(n) + a(n) = A052268(n).

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000007, A010051, A063524.

Join[{0, 1, 1},LinearRecurrence[{1},{0},102]] (* Ray Chandler, Sep 23 2015 *)

a(n)=n==2||n==3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A130130(n) - A130130(n-1), for n>0.

Edited by Charles R Greathouse IV, Mar 23 2010

A021085 Decimal expansion of 1/81.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

The decimal expansion of Sum_{n>=1} floor(n * tanh(Pi))/10^n is the same as that of 1/81 for the first 268 decimal places [Borwein et al.]

Sqrt(999999999999999999) = 9*sqrt(12345679012345679). - Ryohei Miyadera, Ken Hirotomi, Hiroyuki Ozaki and Atushi Tanaka, Jan 16 2006

J. Borwein, D. Bailey and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, Peters, Boston, 2004. See Sect. 1.4.

Jean-François Alcover, 300 digits of Sum_{n>=1} floor(n*tanh(Pi))/10^n
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A052268.

Table[Mod[n, 9], {n, 0, 120}] /. 8 -> 9 (* or *)
PadLeft[First@ #, Abs@ Last@ # + Length@ First@ #] &@ RealDigits[N[1/81, 120]] (* Michael De Vlieger, Jun 21 2016 *)
PadRight[{},120,{0,1,2,3,4,5,6,7,9}](* Harvey P. Dale, Apr 07 2019 *)

Equals Sum_{k >= 1} (1/2^k)*(1/5^k)*k. - Eric Desbiaux, Mar 11 2009
G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 9*x^7)/(1 - x^9). - Ilya Gutkovskiy, Jun 21 2016
From Stefano Spezia, Jun 03 2021: (Start)
a(n) = a(n-9) for n > 8.
Equals (1/10)*Sum_{n>0} 1/A052268(n). (End)

A089186 Decreases from 9 * 10^k down to 1, restarting at 9 * 10^(k+1).

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28

Offset: 1

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Dec 07 2003

Points of local maxima: 9 * 10^k.

a(n) is the 10's complement of n. - Robert Israel, Jul 03 2024

Alois P. Heinz, Table of n, a(n) for n = 1..9999

Cf. A052268, A178914 (essentially the same).

[10^#Intseq(n)-n : n in [1..68]]; // Arkadiusz Wesolowski, Nov 09 2013

a:= n-> 10^length(n)-n:
seq(a(n), n=1..72);  # Alois P. Heinz, Sep 18 2024

Join[Range[9,1,-1],Range[90,1,-1]] (* Harvey P. Dale, Jun 06 2013 *)

a(n) = 10^(1+logint(n, 10)) - n \\ Ruud H.G. van Tol, Jan 16 2023

a(n) = 10^(floor(log_10(n)) + 1) - n.

A333402 Numbers m such that the largest digit in the decimal expansion of 1/m is 1.

Original entry on oeis.org

1, 9, 10, 90, 99, 100, 900, 909, 990, 999, 1000, 9000, 9009, 9090, 9900, 9990, 9999, 10000, 90000, 90009, 90090, 90900, 90909, 99000, 99900, 99990, 99999, 100000, 900000, 900009, 900090, 900900, 909000, 909090, 990000, 990099, 999000, 999900, 999990, 999999, 1000000

Offset: 1

Bernard Schott, Mar 19 2020

If m is a term, 10*m is also a term.
If m is a term then m has only digits {1}, {9}, {1,0} or {9,0} in its decimal representation, but this is not sufficient to be a term (see examples).
Some subsequences below (not exhaustive, see crossrefs):
m = 10^k, k >= 0, hence m is in A011557 = {1, 10, 100, 1000, 10000, ...};
m = 9*10^k, k >= 0, hence m is in A052268 = {9, 90, 900, 9000, 90000, ...};
m = 10^k-1, k >= 1, hence m is in A002283 = {9, 99, 999, 9999, 99999, ...};
m = 9*(10^k+1), k >= 1, hence m is in 9*A000533 = {99, 909, 9009, 90009, ...};
m = 9+100*(100^k-1)/11, k >= 0, hence m is in 9*A094028 = {9, 909, 90909, 9090909, ...}.

			As 1/101 = 0.009900990099..., 101 is not a term.
As 1/909 = 0.001100110011..., 909 is a term.
As 1/9099 = 0.000109902187..., 9099 is not a term.
As 1/9999 = 0.000100010001..., 9999 is also a term.

Index entries for sequences related to decimal expansion of 1/n

Cf. A333236, A333237 (similar, with 9).
Subsequences: A002283, A011557, A052268.
Subsequences: 9*A000533, 9*A094028, 9*A135577, 9*A261544, 9*A330135.

Select[Range[10^4], Max @ RealDigits[1/#][[1]] == 1 &] (* Amiram Eldar, Mar 19 2020 *)

from itertools import count, islice
def A333402_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        k = 1
        while k <= m:
            k *= 10
        rset = {0}
        while True:
            k, r = divmod(k, m)
            if max(str(k)) > '1':
                break
            else:
                if r in rset:
                    yield m
                    break
            rset.add(r)
            k = r
            while k <= m:
                k *= 10
A333402_list = list(islice(A333402_gen(),30)) # Chai Wah Wu, Feb 17 2022

A333236(a(n))= 1.

More terms from Jinyuan Wang, Mar 19 2020

A049415 Number of squares (of positive integers) with n digits.

Original entry on oeis.org

3, 6, 22, 68, 217, 683, 2163, 6837, 21623, 68377, 216228, 683772, 2162278, 6837722, 21622777, 68377223, 216227767, 683772233, 2162277661, 6837722339, 21622776602, 68377223398, 216227766017, 683772233983, 2162277660169

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

a(n) + A180426(n) + A180429(n) + A180347(n) = A052268(n).
Lim_{n->infinity} a(2n)/10^n = 1 - 1/sqrt(10);
lim_{n->infinity} a(2n-1)/10^n = 1/sqrt(10) - 1/10. - Robert G. Wilson v, Aug 29 2012

			22 squares (100=10^2, 121=11^2, ...., 961=31^2) have 3 digits, hence a(3)=22.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

A049415(n) = A017936(n+1) - A017936(n) = A049416(n+1) - A049416(n).

Cf. A062940.

[Ceiling(Sqrt(10^n))-Ceiling(Sqrt(10^(n-1))) : n in [1..30]]; // Vincenzo Librandi, Oct 01 2011

f[n_] := Ceiling[Sqrt[10^n - 1]] - Ceiling[Sqrt[10^(n - 1)]]; f[1] = 3; Array[f, 24] (* Robert G. Wilson v, Aug 29 2012 *)

a(n) = ceiling(sqrt(10^n)) - ceiling(sqrt(10^(n-1))).
From Jon E. Schoenfield, Nov 30 2019: (Start)
a(2n) = floor(10^n * (1 - 1/sqrt(10))), so each even-indexed term a(2n) is given by the first n digits (after the decimal point) of 1 - 1/sqrt(10) = 0.68377223398316...;
a(2n-1) = ceiling(10^n * (1/sqrt(10) - 1/10)), so each odd-indexed term a(2n-1) is given by the first n digits (after the decimal point) of 1/sqrt(10) - 1/10 = 0.21622776601683..., plus 1. (End)

More terms from Dean Hickerson, Jul 10 2001

A100062 Denominator of the probability that an integer n occurs in the cumulative sums of the decimal digits of a random real number between 0 and 1.

Original entry on oeis.org

9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 22876792454961, 205891132094649, 1853020188851841, 16677181699666569, 150094635296999121

Offset: 1

Eric W. Weisstein, Nov 01 2004

Essentially the same as A001019 = powers of 9.

Also number of n-digit positive integers with no identical adjacent digits. Hence the numerator (with A052268 as denominator) of the probability that an n-digit positive integer has this property (e.g., 9/9, 81/90, 729/900, ..., where A100062(n)/A052268(n) reduces to A001019(n-1)/A011557(n-1)). - Rick L. Shepherd, Jun 08 2008

			1/9, 10/81, 100/729, 1000/6561, 10000/59049, ...

Stanislav Sykora, Table of n, a(n) for n = 1..666; previous Table by Rick L. Shepherd went up to n=30.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Evil Number
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019, A011557 A052268, A100061, A100062, A271880, A271881.

[9^n : n in [1..30]]; // Wesley Ivan Hurt, Apr 18 2016

A100062:=n->9^n: seq(A100062(n), n=1..30); # Wesley Ivan Hurt, Apr 18 2016

9^Range[20] (* Harvey P. Dale, Dec 25 2012 *)

\\ The 'old' approach, using the generating function:
s = Vec(Ser((1-x^9)/(x^10-10*x+9),x,666));
a = vector(#s,n,denominator(s[n])) \\ Stanislav Sykora, Apr 16 2016

a(n) = 9^n. - Max Alekseyev, Mar 03 2007
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 9*a(n-1), n>1; a(1)=9.
G.f.: 9x/(1-9x). (End)
a(n) = A001019(n) for n>0. - Wesley Ivan Hurt, Apr 18 2016

More terms from Rick L. Shepherd, Jun 08 2008

A181376 Total number of n-digit numbers requiring 2 positive cubes in their representation as a sum of cubes.

Original entry on oeis.org

2, 7, 32, 161, 736, 3416, 15976, 74295, 345334, 1605089, 7455698, 34623338, 160759047, 746318897, 3464508951, 16081935250, 74648713406

Offset: 1

Martin Renner, Jan 28 2011

A181354(n) + a(n) + A181378(n) + A181380(n) + A181384(n) + A181401(n) + A181403(n) + A181405(n) + A171386(n) = A052268(n).

			a(1) = 2 from 1+1=2, 1+8=9.
a(2) = 7 from 8+8=16, 1+27=28, 35, 54, 65, 72, 91.

Eric W. Weisstein, MathWorld -- Waring's Problem.

Cf. A003325.

Table[Length[c = Table[j^3, {j, (10^n - 1)^(1/3)}];
  Select[Union[Flatten[Outer[Plus, c, c]]],
IntervalMemberQ[Interval[{10^(n - 1), 10^n - 1}], #] &]], {n, 10}] (* Robert Price, Apr 18 2019 *)

a(n)=my(N=10^n, Nn=N/10, v=List(), x3, t); sum(x=sqrtnint(Nn\2,3), sqrtnint(N-1, 3), x3=x^3; sum(y=1, min(sqrtnint(N-x3, 3), x), t=x3+y^3; t>=Nn && !ispower(t, 3) && listput(v, t))); #vecsort(v, , 8) \\ Charles R Greathouse IV, Oct 16 2013

a(n) = A181375(n)-A181375(n-1).

a(6)-a(11) from Charles R Greathouse IV, Oct 16 2013
a(12) from Lars Blomberg, Jan 15 2014
a(13)-a(17) from Hiroaki Yamanouchi, Jul 13 2014

A181378 Total number of n-digit numbers requiring 3 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 14, 107, 1006, 9550, 92743, 913905, 9060358, 90216532

Offset: 1

Martin Renner, Jan 28 2011

A181354(n) + A181376(n) + a(n) + A181380(n) + A181384(n) + A181401(n) + A181403(n) + A181405(n) + A171386(n) = A052268(n)

Eric W. Weisstein, MathWorld -- Waring's Problem.

Cf. A047702

a(n) = A181377(n)-A181377(n-1)

a(5)-a(9) from Lars Blomberg, Jan 15 2014

A181380 Total number of n-digit numbers requiring 4 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 17, 224, 3101, 43340, 558806, 6615757, 73663693, 784419159

Offset: 1

Martin Renner, Jan 28 2011

A181354(n) + A181376(n) + A181378(n) + a(n) + A181384(n) + A181401(n) + A181403(n) + A181405(n) + A171386(n) = A052268(n).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A047703, A181379.

Cf. A052268, A181354, A181376, A181378, A181384, A181401, A181403, A181405, A171386.

a(n) = A181379(n) - A181379(n-1).

a(5)-a(9) from Lars Blomberg, Jan 15 2014

A181384 Total number of n-digit numbers requiring 5 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 20, 272, 3549, 34234, 244503, 1454243, 7201405, 25018440

Offset: 1

Martin Renner, Jan 28 2011

A181354(n) + A181376(n) + A181378(n) + A181380(n) + a(n) + A181401(n) + A181403(n) + A181405(n) + A171386(n) = A052268(n)

Eric W. Weisstein, MathWorld -- Waring's Problem.

Cf. A047704

a(n) = A181381(n)-A181381(n-1)

a(5)-a(9) from Lars Blomberg, Jan 15 2014

cp's OEIS Frontend

A171386 The characteristic function of 2 and 3: 1 if n is prime such that either n-1 or n+1 is prime, else 0.

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A021085 Decimal expansion of 1/81.

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A089186 Decreases from 9 * 10^k down to 1, restarting at 9 * 10^(k+1).

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A333402 Numbers m such that the largest digit in the decimal expansion of 1/m is 1.

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A049415 Number of squares (of positive integers) with n digits.

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A100062 Denominator of the probability that an integer n occurs in the cumulative sums of the decimal digits of a random real number between 0 and 1.

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A181376 Total number of n-digit numbers requiring 2 positive cubes in their representation as a sum of cubes.

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