A155961 -id:A155961 - cp's OEIS frontend

A051302 Numbers whose square can be expressed as the sum of two positive cubes in more than one way.

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

Jud McCranie

Observations regarding terms through a(64)=306761364: All are multiples of 7^2, 13^2, and/or 19^2. Other than 2, 3, 5 and 11, their only prime factors are 7, 13, 19, 31, 43, 61, 67, 79, 127, 151, and 181 (each of which exceeds a multiple of 6 by 1). None is a cube or higher power; the ones that are squares are a(7), a(12), a(17), a(19), a(20), a(32), a(33), a(41), a(49), a(55), and a(58). - Jon E. Schoenfield, Oct 08 2006
Many of the terms beyond a(64) have prime factors other than those found in a(1) through a(64); however, each term through a(774) has at most one distinct prime factor p > 5 that does not exceed a multiple of 6 by 1, and p, if such a prime factor exists, has a multiplicity m=3, with only a few exceptions: n=651 and n=713 (where p^m is 11^2), n=346 and n=770 (where p^m is 17^2), n=699 and n=740 (where p^m is 23^2), and n=741 (where p^m is 11^6). - Jon E. Schoenfield, Oct 20 2013
First differs from A145553 at A051302(172)=3343221000 where 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3.
This sequence is the union of A145553 and A155961.
This sequence is infinite. If n is a member 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, then n*k^3 is in this sequence for all k > 0. - Altug Alkan, May 10 2016

			2989441^2 = 1729^3+20748^3 = 15561^3+17290^3, so 2989441 is in the sequence.

Jon E. Schoenfield and Ray Chandler, Table of n, a(n) for n = 1..774

Cf. A050801, A001235, A011541, A145553, A155961.

(* Warning: this script is only a recomputation of the original b-file of 64 terms from Jon E. Schoenfield, and should not be used to extend the data. *)
max = 310000000; cubeFreeParts = {361, 8281, 33124, 159201, 169309, 221725, 565068, 628849, 917427, 1054729, 2370963, 2989441, 4672423, 8968323, 9402967, 9795747, 34634719};
r[x_] := Reduce[0 < y <= z && x^2 == y^3 + z^3, {y, z}, Integers];
okQ[primes_] := Intersection[{2, 3, 5, 7, 11, 13, 19, 31, 43, 61, 67, 79, 127, 139, 151, 181}, primes] == primes;
crop[n_] := Reap[For[m = 1, True, m++, x = n*m^3; If[x > max, Break[]]; If[okQ[FactorInteger[x][[All, 1]]], If[Head[rx = r[x]] === Or, Print["x = ", x, " ", rx]; Sow[x]];]]][[2, 1]];
A051302 = crop /@ cubeFreeParts // Flatten // Sort (* Jean-François Alcover, Jul 02 2017 *)

T=thueinit('x^3+1, 1);
is(n)=my(v=thue(T, n^2)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1 \\ Charles R Greathouse IV, May 10 2016

Definition corrected by Jon E. Schoenfield, Aug 27 2006
More terms from Jon E. Schoenfield, Oct 08 2006
Extended by Ray Chandler, Nov 22 2011

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

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).

cp's OEIS Frontend

A051302 Numbers whose square can be expressed as the sum of two positive cubes in more than one way.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs