A159843 -id:A159843 - cp's OEIS frontend

A083233 a(n) = (3*8^n + 0^n)/4.

1, 6, 48, 384, 3072, 24576, 196608, 1572864, 12582912, 100663296, 805306368, 6442450944, 51539607552, 412316860416, 3298534883328, 26388279066624, 211106232532992, 1688849860263936, 13510798882111488, 108086391056891904, 864691128455135232

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A083232. Inverse binomial transform of A066443.
Numbers k such that, except for some first term, k^2 = [A000302]^3 + [A004171]^3 + [A002001]^3; e.g., 3072^2 = 64^3 + 128^3 + 192^3; 51539607552^2 = 4194304^3 + 8388608^3 + 12582912^3. - Vincenzo Librandi, Aug 08 2010
With the exception of the first term, these numbers cannot be written as the sum of two integer cubes but can be written as the sum of two positive rational cubes (i.e., 6*8^n = (17*2^n/21)^3 + (37*2^n/21)^3). - Arkadiusz Wesolowski, Aug 15 2013
a(n+1) is the number of unit square faces on the convex hull of a level n Menger sponge. This follows since it has six exterior faces, each of which is a Sierpinski carpet with 8^n squares. - Allan Bickle, Nov 28 2022

			a(0) = (3*8^0 + 0^0)/4 = 4/4 = 1 (using 0^0 = 1).

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A083234. Subsequence of A159843.
Cf. A000302, A004171, A002001.
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

Join[{1},NestList[8#&,6,20]] (* Harvey P. Dale, Sep 25 2020 *)

a(n)=(3*8^n+0^n)/4 \\ Charles R Greathouse IV, Oct 07 2015

def A083233(n): return 3<<3*n-2 if n else 1 # Chai Wah Wu, Nov 27 2023

a(n) = (3*8^n + 0^n)/4.
G.f.: (1-2x)/(1-8x).
E.g.f.: (3*exp(8x) + exp(0))/4.
a(0) = 1, a(n+1) = 6*8^n. - Arkadiusz Wesolowski, Aug 15 2013

A020898 Positive cubefree integers n such that the Diophantine equation X^3 + Y^3 = n*Z^3 has solutions.

Original entry on oeis.org

2, 6, 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 49, 50, 51, 53, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 97, 98, 103, 105, 106, 107, 110, 114, 115, 117, 123, 124, 126, 127, 130

Offset: 1

Steven Finch

These numbers are the cubefree sums of two nonzero rational cubes.
This sequence does not contain A202679, which has members that are not cubefree. - Robert Israel, Mar 16 2016
Notice that 34^3 + 74^3 = 48*21^3 = 6*42^3 because 48 = 6*2^3 is not cubefree, but now 17^3 + 37^3 = 6*21^3 and 6 is already listed in the sequence. - Michael Somos, Mar 13 2023

			37^3 + 17^3 = 6*21^3 is the smallest positive solution for n = 6 (found by Lagrange).
5^3 + 4^3 = 7*3^3 is the smallest positive solution for n = 7.

B. N. Delone and D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Amer. Math. Soc., 1964.
L. E. Dickson, History of The Theory of Numbers, Vol. 2, Chap. XXI, Chelsea NY 1966.
L. J. Mordell, Diophantine Equations, Academic Press, Chap. 15.

David W. Wilson, Table of n, a(n) for n = 1..255 (from Finch paper)
J. H. E. Cohn, The £ 450 question, Math. Mag., 73 (no. 3, 2000), 220-226.
Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]

Cf. A159843, A166246, A228499, A254324, A254326.

(* A naive program with a few pre-computed terms *) nmax = 130; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]} & /@ FactorInteger[n]); nn = Reap[ Do[ n = CubeFreePart[ x*y*(x+y) ]; If[ 1 < n <= nmax, Sow[n]], {x, 1, xmax}, {y, x, xmax}]][[2, 1]] // Union; A020898 = Union[nn, {17, 31, 53, 67, 71, 79, 89, 94, 97, 103, 107, 123}](* Jean-François Alcover, Mar 30 2012 *)

Entry revised by N. J. A. Sloane, Aug 12 2004

Links updated by Max Alekseyev, Oct 17 2007 and Dec 12 2007

A020897 Sum of two nonzero rational cubes.

Original entry on oeis.org

2, 6, 7, 9, 12, 13, 15, 16, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 54, 56, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 110, 114, 115, 117, 120

Offset: 1

Steven Finch

n such that x^3 + y^3 = n*z^3 has a solution in nonnegative integers x,y,z.
Dolan on page 37 states: "Equation 1 is insoluble if n = 31, 38, 67, 76 or 95." - Michael Somos, Nov 18 2021
See the Selmer 1951 article pp. 357-360, Table 6, "The number g of generators and the basic solutions of the equation X^3 + Y^3 = AZ^3, A cubefree and <= 500. - Michael Somos, Feb 15 2022

			6*21^3 = 37^3 + 17^3, 7*3^3 = 5^3 + 4^3, 9*1^3 = 2^3 + 1^3, 12*39^3 = 89^3 + 19^3, 13*3^3 = 7^3 + 2^3, 15*294^3 = 683^3 + 397^3, ... - _Michael Somos_, Nov 18 2021
31*42^3 = 137^3 + (-65)^3, 67*1323^3 = 5353^3 + 1208^3. - _Michael Somos_, Feb 12 2022

S. W. Dolan, On expressing numbers as the sum of two cubes, The Mathematical Gazette, Vol. 66, No. 436 (Mar., 1982), pp. 31-38. See table of solutions on page 32. - _Michael Somos_, Nov 18 2021
Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Ernst S. Selmer, The diophantine equation ax^3 + by^3 + cz^3 = 0, Acta Math. 85 (1951), pp. 203-362.

Subsequence of A159843.

(* A naive program with a few pre-computed terms *) nmax = 120; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]} & /@ FactorInteger[n]); nn = Reap[Do[n = CubeFreePart[x*y*(x+y)]; If[1 < n <= nmax, Sow[n]], {x, 1, xmax}, {y, x, xmax}]][[2, 1]] // Union; A020897 = Select[ Union[nn, nn*2^3, nn*3^3, nn*4^3, {17, 31, 53, 67, 71, 79, 89, 94, 97, 103, 107}], # <= nmax &] (* Jean-François Alcover, Apr 02 2012 *)

Offset corrected by Arkadiusz Wesolowski, Aug 15 2013

A166246 Primes representable as the sum of two rational cubes.

Original entry on oeis.org

2, 7, 13, 17, 19, 31, 37, 43, 53, 61, 67, 71, 79, 89, 97, 103, 107, 127, 139, 151, 157, 163, 179, 193, 197, 211, 223, 229, 233, 241, 251, 269, 271, 277, 283, 313, 331, 337, 349, 359, 367, 373, 379, 397, 409, 421, 431, 433, 439, 449, 457, 463, 467, 499, 503, 521

Offset: 1

Max Alekseyev, Oct 10 2009

nonn

The prime elements of A159843, i.e., the intersection of A159843 and A000040.

Also, the prime elements of A020898.

H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 378.

Fernando Rodriguez Villegas, Don Zagier, Which primes are sums of two cubes?, CMS Conference Proceedings 15 (1995), pp. 295-306.

Cf. A166243, A166244, A159843.

(* To speed up computation, a few terms are pre-computed *) nmax = 521; xmax = 360; preComputed = {127, 271, 379}; solQ[p_] := Do[ If[ IntegerQ[z = Root[-x^3 - y^3 + p*#^3 & , 1]], Print[p, {x, y, z}]; Return[True]], {x, 2, xmax}, {y, x, xmax}]; A166246 = Union[ preComputed, Select[ Prime[ Range[ PrimePi[nmax]]], Mod[#, 9] == 4 || Mod[#, 9] == 7 || Mod[#, 9] == 8 || solQ[#] === True & ]](* Jean-François Alcover, Apr 04 2012, after given formula *)

Under the Birch and Swinnerton-Dyer conjecture, these primes consist of:
(i) p = 2;
(ii) p == 4, 7, or 8 (mod 9);
(iii) p == 1 (mod 9) and p divides A206309(p-1), i.e., Villegas-Zagier polynomial A166243((p-1)/3) evaluated at x=0.

A309960 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 10, 11, 14, 16, 18, 21, 23, 24, 25, 27, 29, 32, 36, 38, 39, 40, 41, 44, 45, 46, 47, 52, 54, 55, 57, 59, 60, 64, 66, 73, 74, 76, 77, 80, 81, 82, 83, 88, 93, 95, 99, 100, 101, 102, 108, 109, 111, 112, 113, 116, 118, 119, 121, 122, 125, 128, 129, 131, 135, 137

Offset: 1

Seiichi Manyama, Aug 25 2019

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Complement of A159843 \ A000578.

Cf. A060748, A060838, A309961 (rank 1), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).

for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==0, print1(k", ")))

is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E, eri, mwr, ar); if(r<6, return(1)); E=ellinit([0, 16*r^2]); eri=ellrankinit(E); mwr=ellrank(eri); if(mwr[1], return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(!ar)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>0, return(0), mwr[2]<1, return(1))); "unknown (0 under BSD conjecture)" \\ Charles R Greathouse IV, Jan 24 2023

A060838(a(n)) = 0.

A309961 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 1.

Original entry on oeis.org

6, 7, 9, 12, 13, 15, 17, 20, 22, 26, 28, 31, 33, 34, 35, 42, 43, 48, 49, 50, 51, 53, 56, 58, 61, 62, 63, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 87, 89, 90, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 114, 115, 117, 120, 123, 130, 133, 134, 136, 139, 140, 141, 142, 143, 151

Offset: 1

Seiichi Manyama, Aug 25 2019

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A159843.

Cf. A060748, A060838, A309960 (rank 0), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).

for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==1, print1(k", ")))

is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E=ellinit([0, 16*r^2]), eri=ellrankinit(E), mwr=ellrank(eri), ar); if(r<6 || mwr[1]==0, return(0)); if(mwr[2]>1, return(0)); ar=ellanalyticrank(E)[1]; if(ar==0, return(0)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>1 || mwr[2]<1, return(0), mwr[1]==mwr[2] && mwr[1]==1, return(1))); error("unknown (",ar==1," on the BSD conjecture)") \\ Charles R Greathouse IV, Jan 24 2023

A060838(a(n)) = 1.

A309962 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 2.

Original entry on oeis.org

19, 30, 37, 65, 86, 91, 110, 124, 126, 127, 132, 152, 153, 163, 182, 183, 201, 203, 209, 210, 217, 218, 219, 240, 246, 254, 271, 273, 282, 296, 309, 335, 342, 345, 348, 370, 379, 390, 397, 399, 407, 420, 433, 435, 436, 446, 453, 462, 468, 469, 477, 497, 498, 506, 513, 520, 523, 554

Offset: 1

Seiichi Manyama, Aug 25 2019

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A159843.

Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309963 (rank 3), A309964 (rank 4).

for(k=1, 1e3, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==2, print1(k", ")))

A060838(a(n)) = 2.

A309963 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 3.

Original entry on oeis.org

657, 854, 1020, 1122, 1241, 1267, 1330, 1339, 1426, 1482, 1554, 1798, 1853, 1892, 2015, 2310, 2346, 2574, 2763, 2771, 2805, 2869, 2914, 2947, 2977, 3036, 3383, 3445, 3465, 3526, 3894, 3913, 4002, 4209, 4290, 4362, 4706, 4711, 4830, 4921, 4930, 4977, 5025, 5053, 5074, 5193, 5256

Offset: 1

Seiichi Manyama, Aug 25 2019

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A159843.

Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309962 (rank 2), A309964 (rank 4).

for(k=1, 5e3, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==3, print1(k", ")))

A060838(a(n)) = 3.

A293648 Sum of two (possibly negative) coprime cubes in at least 2 ways, but not the sum of 2 noncoprime cubes.

Original entry on oeis.org

91, 217, 721, 1027, 1729, 3367, 4706, 4921, 4977, 7657, 8587, 8911, 9919, 10621, 14911, 15561, 16263, 20683, 21014, 23877, 25669, 27937, 28063, 31519, 35929, 39331, 40033, 49959, 63693, 68705, 68857, 68913, 73017, 77653, 77779, 97309, 98623, 106597, 109573

Offset: 1

Rosalie Fay, Oct 16 2017

nonn

Not every term is cubefree; some are sb^3 where s is in A159843 and b > 1.

			63693 = 34^3 + 29^3 = 53^3 - 44^3 and 34 & 29 are coprime, as are 53 & -44, but it is not also the sum of cubes of 2 noncoprime integers, so 63693 is in the sequence.

Rosalie Fay, Table of n, a(n) for n = 1..325

Cf. A293647 (allows noncoprime); A293649 (only positives); A293646, A293651

A309964 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 4.

Original entry on oeis.org

21691, 27937, 33193, 34706, 36667, 39331, 45353, 46299, 53265, 55298, 55335, 59295, 59690, 62628, 63147, 64001, 65683, 73963, 78604, 82290, 87653, 90489, 94681, 96139

Offset: 1

Seiichi Manyama, Aug 25 2019

nonn

Subsequence of A159843.

Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309962 (rank 2), A309963 (rank 3).

for(k=1, 5e4, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==4, print1(k", ")))

A060838(a(n)) = 4.

a(18)-a(24) from Maksym Voznyy, Jan 25 2023

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A083233 a(n) = (3*8^n + 0^n)/4.

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A020898 Positive cubefree integers n such that the Diophantine equation X^3 + Y^3 = n*Z^3 has solutions.

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A020897 Sum of two nonzero rational cubes.

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A166246 Primes representable as the sum of two rational cubes.

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A309960 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 0.

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A309961 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 1.

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A309962 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 2.

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A309963 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 3.

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