A061439 -id:A061439 - cp's OEIS frontend

A018005 Smallest number whose cube has n digits.

1, 3, 5, 10, 22, 47, 100, 216, 465, 1000, 2155, 4642, 10000, 21545, 46416, 100000, 215444, 464159, 1000000, 2154435, 4641589, 10000000, 21544347, 46415889, 100000000, 215443470, 464158884, 1000000000, 2154434691, 4641588834

Offset: 1

N. J. A. Sloane

With offset 0, ((cube root of 10) to the power n) rounded up.
From Carmine Suriano, Mar 14 2020: (Start)
The terms corresponding to n = (20,21); (38,39); (41,42); (56,57); (59,60); (77,78); (80,81) ... are such that the square of first term starts with the digits of second term, and the square of second term starts with the digits of the first. For example, a(38)^2 = 2154434690032^2 = 4641588833613.... and a(39)^2 = 4641588833613^2 = 2154434690032...
 (End)

			a(5) = 22, 22^3 = 10648 has 5 digits, while 21^3 = 9261 has 4 digits.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A061434, A061439, and powers of cube root of k ceiling up: A017981 (k=2), A017984 (k=3), A017987 (k=4), A017990 (k=5), A017993 (k=6), A017996 (k=7), A018002 (k=9), this sequence (k=10), A018008 (k=11), A018011 (k=12), A018014 (k=13), A018017 (k=14), A018020 (k=15), A018023 (k=16), A018026 (k=17), A018029 (k=18), A018032 (k=19), A018035 (k=20), A018038 (k=21), A018041 (k=22), A018044 (k=23), A018047 (k=24).

[Ceiling(10^(n/3)): n in [0..40]]; // Vincenzo Librandi, Jan 09 2014

Table[Ceiling[10^(n/3)], {n, 0, 40}] (* Vincenzo Librandi, Jan 09 2014 *)

More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001

A130130 a(0)=0, a(1)=1, a(n)=2 for n >= 2.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Aug 01 2007

a(n) is also total number of positive integers below 10^(n+1) requiring 9 positive cubes in their representation as sum of cubes (cf. Dickson, 1939).
A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + a(n) = A002283(n).
a(n) = number of obvious divisors of n. The obvious divisors of n are the numbers 1 and n. - Jaroslav Krizek, Mar 02 2009
Number of colors needed to paint n adjacent segments on a line. - Jaume Oliver Lafont, Mar 20 2009
a(n) = ceiling(n-th nonprimes/n) = ceiling(A018252(n)/A000027(n)) for n >= 1. Numerators of (A018252(n)/A000027(n)) in A171529(n), denominators of (A018252(n)/A000027(n)) in A171530(n). a(n) = A171624(n) + 1 for n >= 5. - Jaroslav Krizek, Dec 13 2009
a(n) is also the continued fraction for sqrt(1/2). - Enrique Pérez Herrero, Jul 12 2010
For n >= 1, a(n) = minimal number of divisors of any n-digit number. See A066150 for maximal number of divisors of any n-digit number. - Jaroslav Krizek, Jul 18 2010
Central terms in the triangle A051010. - Reinhard Zumkeller, Jun 27 2013
Decimal expansion of 11/900. - Elmo R. Oliveira, May 05 2024

Leonard Eugene Dickson, All integers except 23 and 239 are sums of eight cubes, Bulletin of the American Mathematical Society 45 (1939), p. 588-591.
Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000005, A000027, A002283, A018252, A051010, A061439, A066150, A070824, A158411, A171529, A171530, A171624, A181375, A181377, A181379, A181381, A181400, A181402, A181404.

a130130 = min 2
a130130_list = 0 : 1 : repeat 2  -- Reinhard Zumkeller, Jun 27 2013

A130130[0]:=0; A130130[1]:=1; A130130[n_]:=2; (* Enrique Pérez Herrero, Jul 12 2010 *)
A130130[n_]:=ContinuedFraction[Sqrt[1/2],n+1][[n+1]] (* Enrique Pérez Herrero, Jul 12 2010 *)
Join[{0, 1},LinearRecurrence[{1},{2},96]] (* Ray Chandler, Sep 23 2015 *)
PadRight[{0,1},120,{2}] (* Harvey P. Dale, Sep 15 2022 *)

a(n)=min(n,2) \\ Charles R Greathouse IV, Jun 01 2011

G.f.: x*(1+x)/(1-x) = x*(1-x^2)/(1-x)^2. - Jaume Oliver Lafont, Mar 20 2009
a(n) = A000005(n) - A070824(n) for n >= 1.
E.g.f.: 2*exp(x) - x - 2. - Stefano Spezia, May 19 2024

A114322 Largest number whose 4th power has n digits.

Original entry on oeis.org

1, 3, 5, 9, 17, 31, 56, 99, 177, 316, 562, 999, 1778, 3162, 5623, 9999, 17782, 31622, 56234, 99999, 177827, 316227, 562341, 999999, 1778279, 3162277, 5623413, 9999999, 17782794, 31622776, 56234132, 99999999, 177827941, 316227766, 562341325, 999999999, 1778279410

Offset: 1

Jonathan Vos Post, Feb 06 2006

This is to 4th powers as A061439 is to cubes and A049416 is to squares.

a(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

			a(10) = 316 because 316^4 = 9971220736 which has 10 digits, while 317^4 = 10098039121 has 11 digits.
a(35) = 562341325 because 562341325^4 = 99999999864602459914272843469140625 has 35 digits, while 562341326^4 = 100000000575914225104884587789852176 has 36.

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

Cf. A061439, A049416.

[Ceiling((10^n)^(1/4))-1: n in [1..40]]; // Vincenzo Librandi, Oct 01 2011

Ceiling[(10^Range[50])^(1/4)] - 1 (* Paolo Xausa, Jul 30 2024 *)

a(n) = ceiling((10^n)^(1/4)) - 1.

A181354 Number of n-digit perfect cubes.

Original entry on oeis.org

2, 2, 5, 12, 25, 53, 116, 249, 535, 1155, 2487, 5358, 11545, 24871, 53584, 115444, 248715, 535841, 1154435, 2487154, 5358411, 11544347, 24871542, 53584111, 115443470, 248715414, 535841116, 1154434691, 2487154143, 5358411166

Offset: 1

Martin Renner, Jan 28 2011

a(n) is also the total number of n-digit numbers requiring 1 positive cube in their representation as sum of cubes.
a(n) + A181376(n) + A181378(n) + A181380(n) + A181384(n) + A181401(n) + A181403(n) + A181405(n) + A171386(n) = A052268(n).
Differs from A062941 only at n=1, because 0 is considered a 0-digit, not a 1-digit number here. - R. J. Mathar, Jul 09 2011

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

Cf. A000578

a:=n->ceil(10^(n/3))-ceil(10^((n-1)/3));

With[{c = Range[4650000]^3}, Length[#]&/@Table[Select[c, IntegerLength[#] == n &], {n, 20}]] (* Harvey P. Dale, Feb 01 2011 *)
Differences[Ceiling[10^(Range[0, 30]/3)]]

a(n) = A061439(n) - A061439(n-1).

More terms from T. D. Noe, Feb 01 2011

A181404 Total number of positive integers below 10^n requiring 8 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

0, 3, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Martin Renner, Jan 28 2011

Also continued fraction expansion of (9+sqrt(229))/74. - Bruno Berselli, Sep 09 2011

Eric Weisstein's World of Mathematics, Waring's Problem
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A018889, A010854.

Cf. A002283, A061439, A130130, A181375, A181377, A181379, A181381, A181400, A181402.

PadRight[{0, 3}, 100, 15] (* Paolo Xausa, May 24 2024 *)

a(n)=if(n>2,15,3*n-3) \\ Charles R Greathouse IV, Oct 07 2015

A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + a(n) + A130130(n) = A002283(n).
a(n) = 15 for n > 2. - Charles R Greathouse IV, Sep 09 2011
G.f.: 3*x^2*(1+4*x)/(1-x). - Bruno Berselli, Sep 09 2011
E.g.f.: 3*(5*(exp(x) - 1 - x) - 2*x^2). - Stefano Spezia, May 21 2024

a(5)-a(7) from Lars Blomberg, May 04 2011

A181375 Total number of positive integers below 10^n requiring 2 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

2, 9, 41, 202, 938, 4354, 20330, 94625, 439959, 2045048, 9500746, 44124084, 204883131, 951202028, 4415710979, 20497646229, 95146359635

Offset: 1

Martin Renner, Jan 28 2011

A061439(n) + a(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n).

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

Cf. A003325.

iscube:=proc(n) if root(n,3)=trunc(root(n,3)) then true; else false; fi; end:
isA003325:=proc(n) local x,y3; if iscube(n) then false; else for x from 1 do y3:=n-x^3; if y3A003325(k) then i:=i+1; fi; od: return(i); end:
for n from 1 do print(a(n)); od;

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

a(6)-a(12) from Lars Blomberg, May 04 2011

a(13)-a(17) from Hiroaki Yamanouchi, Jul 12 2014

A181377 Total number of positive integers below 10^n requiring 3 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 15, 122, 1128, 10678, 103421, 1017326, 10077684, 100294216

Offset: 1

Martin Renner, Jan 28 2011

A061439(n) + A181375(n) + a(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n).

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

Cf. A047702.

a(5)-a(9) from Lars Blomberg, May 04 2011

A181379 Total number of positive integers below 10^n requiring 4 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 18, 242, 3343, 46683, 605489, 7221246, 80884939, 865304098

Offset: 1

Martin Renner, Jan 28 2011

A061439(n) + A181375(n) + A181377(n) + a(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n).

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

Cf. A047703.

a(5)-a(9) from Lars Blomberg, May 04 2011

A181381 Total number of positive integers below 10^n requiring 5 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 21, 293, 3842, 38076, 282579, 1736822, 8938227, 33956667

Offset: 1

Martin Renner, Jan 28 2011

nonn

A061439(n) + A181375(n) + A181377(n) + A181379(n) + a(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n)

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

Cf. A047704.

a(5)-a(7) from Lars Blomberg, May 04 2011

a(8)-a(9) from Hiroaki Yamanouchi, Sep 23 2014

A181400 Total number of positive integers below 10^n requiring 6 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 18, 202, 1325, 3440, 3919, 3922, 3922, 3922

Offset: 1

Martin Renner, Jan 28 2011

A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + a(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n)

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

Cf. A046040.

a(5)-a(7) from Lars Blomberg, May 04 2011

a(8)-a(9) from Hiroaki Yamanouchi, Sep 23 2014

cp's OEIS Frontend

A018005 Smallest number whose cube has n digits.

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A130130 a(0)=0, a(1)=1, a(n)=2 for n >= 2.

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A114322 Largest number whose 4th power has n digits.

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A181354 Number of n-digit perfect cubes.

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A181404 Total number of positive integers below 10^n requiring 8 positive cubes in their representation as sum of cubes.

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A181375 Total number of positive integers below 10^n requiring 2 positive cubes in their representation as sum of cubes.

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A181377 Total number of positive integers below 10^n requiring 3 positive cubes in their representation as sum of cubes.

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