A181381 -id:A181381 - cp's OEIS frontend

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

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

A061439 Largest number whose cube has n digits.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, May 03 2001

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

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

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

Digits := 100:
A061439 := n->ceil(10^(n/3))-1:
seq (A061439(n), n=1..40);

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 *)

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

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

Typo in Maple program fixed by Martin Renner, Jan 31 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

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

A181402 Total number of positive integers below 10^n requiring 7 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

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

Offset: 1

Martin Renner, Jan 28 2011

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

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.

Cf. A018890.

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

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

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

a(8)-a(34) from Charles R Greathouse IV, Jan 23 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

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

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A061439 Largest number whose cube has n digits.

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

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

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A181402 Total number of positive integers below 10^n requiring 7 positive cubes in their representation as sum of cubes.

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