A051302 -id:A051302 - cp's OEIS frontend

A155961 Numbers whose square can be expressed as the sum of two positive cubes in at least 3 ways.

3343221000, 26745768000, 90266967000

Offset: 1

Ray Chandler, Jan 31 2009

Although this sequence has keyword "bref", this sequence is infinite since if n is in this sequence, then n*k^3 is in this sequence for all k > 0. - Altug Alkan, May 10 2016

			a(1)=3343221000 where 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3.

Uwe Hollerbach, Taxi, Taxi! [Original link, broken]
Uwe Hollerbach, Taxi, Taxi! [Replacement link to Wayback Machine]
Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]

Cf. A051302, A155960.

a(n) = sqrt(A155960(n)).

A217248 Numbers whose square is the sum of two nonnegative cubes.

Original entry on oeis.org

0, 1, 3, 4, 8, 24, 27, 32, 64, 81, 98, 108, 125, 168, 192, 216, 228, 256, 312, 343, 375, 500, 512, 525, 588, 648, 671, 729, 784, 847, 864, 1000, 1014, 1029, 1183, 1225, 1261, 1323, 1331, 1344, 1372, 1536, 1728, 1824, 2048, 2187, 2197, 2496, 2646, 2744, 2888

Offset: 1

Christian N. K. Anderson & Kevin L. Schwartz, Mar 20 2013

nonn

Numbers N such that N^2 = x^3 + y^3 where x and y are nonnegative integers. First case with 2 solutions is 77976^2 = 228^3 + 1824^3 = 1026^3 + 1710^3, see A051302. - Zak Seidov, Mar 21 2013

			312 is in the sequence because 312^2 = 2^3 + 46^3.

Chai Wah Wu, Table of n, a(n) for n = 1..5000

This sequence with only positive (nonzero) cubes: A050801, and that sequence squared: A050802

A natural extension of the hypotenuse numbers A009003.

m = 2888; Sort[Reap[Do[If[IntegerQ[c = Sqrt[a^3 + b^3]], Sow[c]], {a, 0, m^(2/3)}, {b, a, (m^2 - a^3)^(1/3)}]][[2, 1]]] (* Zak Seidov, Mar 21 2013 *)

is(n)=n*=n;for(k=ceil((n/2-.5)^(1/3)),(n+.5)^(1/3),if(ispower(n-k^3,3),return(1)));0 \\ Charles R Greathouse IV, Mar 20 2013

y=c(); maxsol=3000 #All solutions x)==as.integer(x))y=c(y,x)
sort(y)

Offset and a(35) corrected and a(36)-a(51) from Giovanni Resta, Mar 20 2013

A145553 Numbers n such that n^2 can be expressed as the sum of 2 positive cubes in exactly 2 different ways.

Original entry on oeis.org

77976, 223587, 623808, 894348, 1788696, 2105352, 2989441, 4298427, 4672423, 4990464, 5986575, 6036849, 7154784, 8437832, 9747000, 14309568, 16842816, 23915528, 24147396, 24770529, 26745768, 27948375, 34387416, 34634719, 36570744, 37379384, 39923712, 47892600

Offset: 1

Iain Renfrew (iain.renfrew(AT)btinternet.com), Oct 13 2008

nonn

This is conjectured to be an infinite sequence.
Subsequence of A051302. [R. J. Mathar, Oct 14 2008]
First differs from A051302 at A051302(172)=3343221000 where 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3.
If n is a term of this sequence, then n^2 = a^3 + b^3 = c^3 + d^3 where (a, b) and (c, d) are distinct pairs. If n^2 = a^3 + b^3 = c^3 + d^3, then (n*k^3)^2 = n^2*k^6 = k^6*(a^3 + b^3) = k^6*(c^3 + d^3) = (a*k^2)^3 + (b*k^2)^3 = (c*k^2)^3 + (d*k^2)^3. It is obvious that if (a, b) and (c, d) are distinct, then (k^2*a, k^2*b), (k^2*c, k^2*d) are also distinct for all nonzero values of k. So if n is in this sequence and n*k^3 is not in A155961, then n*k^3 is in this sequence for all k > 0. If this sequence is not infinite, then there are infinitely many consecutive k values for any term n such that n*k^3 is in A155961. Is it possible? - Altug Alkan, May 10 2016

			a(1): 77976^2 = 6080256576 = 1824^3 + 228^3 = 1710^3 + 1026^3;
a(2): 223587^2 = 49991146569 = 3666^3 + 897^3 = 3276^3 + 2457^3;
a(3): 623808^2 = 389136420864 = 7296^3 + 912^3 = 6840^3 + 4104^3;
a(4): 894348^2 = 799858345104 = 9282^3 + 546^3 = 9009^3 + 4095^3.

Ray Chandler, Table of n, a(n) for n = 1..771

Cf. A145552, A051302.

a(5)-a(15) from Zak Seidov, Oct 15 2008

Extended by Ray Chandler, Nov 22 2011

A155960 Squares which can be expressed as the sum of two positive cubes in at least 3 ways.

Original entry on oeis.org

11177126654841000000, 715336105909824000000, 8148125331379089000000

Offset: 1

Ray Chandler, Jan 31 2009

			a(1) = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3. - _Jean-François Alcover_, Jul 03 2017

Uwe Hollerbach, Taxi, Taxi! [Original link, broken]
Uwe Hollerbach, Taxi, Taxi! [Replacement link to Wayback Machine]
Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]

Cf. A051302, A155961.

a(n) = A155961(n)^2.

A230719 Smallest number whose square is the sum of two positive cubes in at least n ways.

Original entry on oeis.org

3, 77976, 3343221000

Offset: 1

Jonathan Sondow, Oct 28 2013

See A050801, A051302, A155961 for more comments, references, links, and crossrefs.

			3^2 = 1^3 + 2^3.
77976^2 = 228^3 + 1824^3 = 1026^3 + 1710^3.
3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3.

Index to sequences related to sums of cubes

a(1) = A050801(1), a(2) = A051302(1), a(3) = A155961(1).

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A155961 Numbers whose square can be expressed as the sum of two positive cubes in at least 3 ways.

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A217248 Numbers whose square is the sum of two nonnegative cubes.

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A145553 Numbers n such that n^2 can be expressed as the sum of 2 positive cubes in exactly 2 different ways.

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A155960 Squares which can be expressed as the sum of two positive cubes in at least 3 ways.

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Formula

A230719 Smallest number whose square is the sum of two positive cubes in at least n ways.

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