A181381 -id:A181381

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A130130

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

+10
24

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..97.

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).

FORMULA

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

MATHEMATICA

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 *)

PROG

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

(Haskell)

a130130 = min 2

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Aug 01 2007

STATUS

approved

A061439

Largest number whose cube has n digits.

+10
15

2, 4, 9, 21, 46, 99, 215, 464, 999, 2154, 4641, 9999, 21544, 46415, 99999, 215443, 464158, 999999, 2154434, 4641588, 9999999, 21544346, 46415888, 99999999, 215443469, 464158883, 999999999, 2154434690, 4641588833, 9999999999

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n).

LINKS

Table of n, a(n) for n=1..30.

FORMULA

a(n) = ceiling(10^(n/3)) - 1. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 30 2003

EXAMPLE

a(5) = 46 because 46^3 = 97336 has 5 digits, while 47^3 = 103823 has 6 digits.

MAPLE

Digits := 100:

A061439 := n->ceil(10^(n/3))-1:

seq (A061439(n), n=1..40);

MATHEMATICA

t={}; i=0; Do[i=i+1; While[IntegerLength[i^3]<=n, i++]; AppendTo[t, i-1], {n, 20}]; t (* Jayanta Basu, May 19 2013 *)

CROSSREFS

a(n) is one more than the corresponding term of A018005. Cf. A061435.

KEYWORD

nonn,base,easy

AUTHOR

Amarnath Murthy, May 03 2001

EXTENSIONS

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

Typo in Maple program fixed by Martin Renner, Jan 31 2011

STATUS

approved

A181404

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..61.

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

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

FORMULA

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A018889, A010854.

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

KEYWORD

nonn,easy

AUTHOR

Martin Renner, Jan 28 2011

EXTENSIONS

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

STATUS

approved

A181375

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..17.

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

MAPLE

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 y3<x^3 then return false; elif iscube(y3) then return true; fi; od; fi; end:

a:=proc(n) local i, k; i:=0; for k from 2 to 10^n-1 do if isA003325(k) then i:=i+1; fi; od: return(i); end:

for n from 1 do print(a(n)); od;

PROG

(PARI) 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

CROSSREFS

Cf. A003325.

KEYWORD

nonn,more

AUTHOR

Martin Renner, Jan 28 2011

EXTENSIONS

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

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

STATUS

approved

A181377

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..9.

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

CROSSREFS

Cf. A047702.

KEYWORD

nonn,more

AUTHOR

Martin Renner, Jan 28 2011

EXTENSIONS

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

STATUS

approved

A181379

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..9.

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

CROSSREFS

Cf. A047703.

KEYWORD

nonn,more

AUTHOR

Martin Renner, Jan 28 2011

EXTENSIONS

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

STATUS

approved

A181400

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..9.

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

CROSSREFS

Cf. A046040.

KEYWORD

nonn,more

AUTHOR

Martin Renner, Jan 28 2011

EXTENSIONS

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

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

STATUS

approved

A181402

Total number of positive integers below 10^n requiring 7 positive cubes in their representation as sum of cubes.

+10
9

1, 10, 73, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

An unpublished result of Deshouillers-Hennecart-Landreau, combined with Lemma 3 from Bertault, Ramaré, & Zimmermann implies that a(4)-a(34) are all 121. Probably a(n) = 121 for all n > 3. - Charles R Greathouse IV, Jan 23 2014

LINKS

Table of n, a(n) for n=1..34.

F. Bertault, O. Ramaré, and P. Zimmermann, On sums of seven cubes, Math. Comp. 68 (1999), pp. 1303-1310.

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

FORMULA

A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + a(n) + A181404(n) + A130130(n) = A002283(n).

Conjectured g.f.: x*(1+9*x+63*x^2+48*x^3)/(1-x). - Colin Barker, May 04 2012

Conjectured e.g.f.: 121*(exp(x) - 1) - 120*x - 111*x^2/2 - 8*x^3. - Stefano Spezia, May 21 2024

CROSSREFS

Cf. A018890.

Cf. A002283, A061439, A130130, A181375, A181377, A181379, A181381, A181400, A181404.

KEYWORD