A001014 -id:A001014 - cp's OEIS frontend

A001016 Eighth powers: a(n) = n^8.

0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, 4294967296, 6975757441, 11019960576, 16983563041, 25600000000, 37822859361, 54875873536, 78310985281, 110075314176

Offset: 0

N. J. A. Sloane

Besides the first term, this sequence lists the denominators in Pi^8/9450 = 1 + 1/256 + 1/6561 + 1/65536 + 1/390625 + 1/1679616 + ... - Mohammad K. Azarian, Nov 01 2011, edited by M. F. Hasler, Jul 03 2025
For n > 0, a(n) is the largest number k such that k + n^4 divides k^2 + n^4. - Derek Orr, Oct 01 2014
Fourth powers of squares and squares of 4th powers. Squares composed with themselves twice. - Wesley Ivan Hurt, Apr 01 2016

Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A000290 (squares), A000583 (fourth powers), A001014 - A001017 (6th - 9th powers), A008454 (10th powers), A010801 (13th powers).
Cf. A000542 (partial sums), A022524 (first differences), A013666 (zeta(8)).
Cf. A003380 - A003390 (sums of 2, ..., 12 eighth powers).

[n^8 : n in [0..50]]; // Wesley Ivan Hurt, Apr 01 2016

A001016:=n->n^8: seq(A001016(n), n=0..50); # Wesley Ivan Hurt, Apr 01 2016

Table[n^8, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)

A001016(n):=n^8$
makelist(A001016(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

A001016(n)=n^8 \\ Charles R Greathouse IV, Sep 24 2015

A001016 = lambda n: n**8 # M. F. Hasler, Jul 03 2025

Multiplicative with a(p^e) = p^(8e). - David W. Wilson, Aug 01 2001
Totally multiplicative sequence with a(p) = p^8 for primes p. - Jaroslav Krizek, Nov 01 2009
G.f.: -x*(1+x)*(x^6+246*x^5+4047*x^4+11572*x^3+4047*x^2+246*x+1)/(x-1)^9. - R. J. Mathar, Jan 07 2011
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) + 40320. - Ant King, Sep 24 2013
From Wesley Ivan Hurt, Apr 01 2016: (Start)
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n > 8.
a(n) = A000290(n)^4 = A000290(A000290(A000290(n))).
a(n) = A000583(n)^2. (End)
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(8) = Pi^8/9450 (A013666).
Sum_{n>=1} (-1)^(n+1)/a(n) = 127*zeta(8)/128 = 127*Pi^8/1209600. (End)
E.g.f.: exp(x)*x*(1 + 127*x + 966*x^2 + 1701*x^3 + 1050*x^4 + 266*x^5 + 28*x^6 + x^7). - Stefano Spezia, Jul 29 2022

More terms from James Sellers, Sep 19 2000

A081119 Number of integral solutions to Mordell's equation y^2 = x^3 + n.

Original entry on oeis.org

5, 2, 2, 2, 2, 0, 0, 7, 10, 2, 0, 4, 0, 0, 4, 2, 16, 2, 2, 0, 0, 2, 0, 8, 2, 2, 1, 4, 0, 2, 2, 0, 2, 0, 2, 8, 6, 2, 0, 2, 2, 0, 2, 4, 0, 0, 0, 2, 2, 2, 0, 2, 0, 2, 2, 2, 6, 0, 0, 0, 0, 0, 4, 5, 8, 0, 0, 4, 0, 0, 2, 2, 12, 0, 0, 2, 0, 0, 2, 8, 2, 2, 0, 0, 0, 0, 0, 0, 8, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 12

Offset: 1

T. D. Noe, Mar 06 2003

Mordell's equation has a finite number of integral solutions for all nonzero n.
Gebel, Petho, and Zimmer (1998) computed the solutions for |n| <= 10^4. Bennett and Ghadermarzi (2015) extended this bound to |n| <= 10^7.
Sequence A054504 gives n for which there are no integral solutions. See A081120 for the number of integral solutions to y^2 = x^3 - n.
a(n) is odd iff n is a cube. - Bernard Schott, Nov 23 2019
From Jianing Song, Aug 24 2022: (Start)
a(n) = 5 if n is a sixth power. Further more, if A060950(n) = 0 (namely the elliptic curve y^2 = x^3 + n has rank 0), then:
- a(n) = 2 if n is a square but not a sixth power;
- a(n) = 1 if n is a cube but not a sixth power;
- a(n) = 0 otherwise.
This follows from the complete description of the torsion group of y^2 = x^3 + n, using O to denote the point at infinity (see Exercise 10.19 of Chapter X of Silverman's Arithmetic of elliptic curves):
- If n = t^6 is a sixth power, then the torsion group consists of O, (2*t^2,+-3*t^3), (0,+-t^3), and (-t^2, 0).
- If n = t^2 is not a sixth power, then the torsion group consists of O and (0,+-t).
- If n = t^3 is not a sixth power, then the torsion group consists of O and (-t,0).
- If n is of the form -432*t^6, then the torsion group consists of O and (12*t^2,+-36*t^3).
- In all the other cases, the torsion group is trivial.
So a torsion point on y^2 = x^3 + n other than O is an integral point. If y^2 = x^3 + n has rank 0, then all the integral points on y^2 = x^3 + n are exactly the torsion points other than O.
Note that this result implies particularly that a(n) = a(n*t^6) for all t if A060950(n) = 0: the elliptic curve y^2 = x^3 + n*t^6 can be written as (y/t^3)^2 = (x/t^2)^3 + n, so it has the same Mordell-Weil group (hence the same rank and isomorphic torsion group) as y^2 = x^3 + n. (End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367.

Jean-François Alcover, Table of n, a(n) for n = 1..10000 [There were errors in the previous b-file, which had 10000 terms contributed by T. D. Noe and was based on the work of J. Gebel.]
Michael A. Bennett, Amir Ghadermarzi Mordell's equation: a classical approach. LMS J. Compute. Math. 18 (2015): 633-646. doi:10.1112/S1461157015000182 arXiv:1311.7077
J. Gebel, Integer points on Mordell curves, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
J. Gebel, A. Pethö and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367. (doi:10.1023/A:1000281602647 not working as of July 2024.)
Joseph H. Silverman, The Arithmetic of Elliptic Curves.
Eric Weisstein's World of Mathematics, Mordell Curve.
Wikipedia, Mordell curve.

Cf. A054504, A081120. See A134108 for another version.

(* This naive approach gives correct results up to n = 1000 *) xmax[] = 10^4; Do[xmax[n] = 10^5, {n, {297, 377, 427, 885, 899}}]; Do[xmax[n] = 10^6, {n, {225, 353, 618 }}]; f[n] := (x = -Ceiling[n^(1/3)]-1; s = {}; While[x <= xmax[n], x++; y2 = x^3 + n; If[y2 >= 0, y = Sqrt[y2]; If[ IntegerQ[y], AppendTo[s, y]]]]; s); a[n_] := (fn = f[n];  If[fn == {}, 0, 2 Length[fn] - If[First[fn] == 0, 1, 0] ]); Table[an = a[n]; Print["a[", n, "] = ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Oct 18 2011 *)

Edited by Max Alekseyev, Feb 06 2021

A054504 Numbers n such that Mordell's equation y^2 = x^3 + n has no integral solutions.

Original entry on oeis.org

6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42, 45, 46, 47, 51, 53, 58, 59, 60, 61, 62, 66, 67, 69, 70, 74, 75, 77, 78, 83, 84, 85, 86, 87, 88, 90, 93, 95, 96, 102, 103, 104, 109, 110, 111, 114, 115, 116, 118, 123, 124, 130, 133, 135, 137, 139, 140, 146, 147, 149, 153, 155

Offset: 1

N. J. A. Sloane, Apr 08 2000

Mordell's equation has a finite number of integral solutions for all nonzero n. Gebel computes the solutions for n < 10^5. Sequence A081121 gives n for which there are no integral solutions to y^2 = x^3 - n. See A081119 for the number of integral solutions to y^2 = x^3 + n. - T. D. Noe, Mar 06 2003

Numbers n such that A081119(n) = 0. - Charles R Greathouse IV, Apr 29 2015

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 192.
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367.

T. D. Noe, Table of n, a(n) for n = 1..6603 (from Gebel)
Pantelis Andreou, Stavros Konstantinidis, and Taylor J. Smith, Improved Randomized Approximation of Hard Universality and Emptiness Problems, arXiv:2403.08707 [cs.DS], 2024. See p. 16.
Ryan D'Mello, Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves, arXiv preprint arXiv:1410.0078, 2014
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Eric Weisstein's World of Mathematics, Mordell Curve

Cf. A081119, A081121.

m = 155; f[_List] := ( xm = 2 xm; ym = Ceiling[xm^(3/2)];
Complement[Range[m], Outer[Plus, Range[0, ym]^2, -Range[-xm, xm]^3] //Flatten //Union]); xm=10; FixedPoint[f, {}] (* Jean-François Alcover, Apr 28 2011 *)

Apostol gives all values of n < 100. Extended by David W. Wilson, Sep 25 2000

A008455 11th powers: a(n) = n^11.

Original entry on oeis.org

0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, 1792160394037, 4049565169664, 8649755859375, 17592186044416, 34271896307633, 64268410079232, 116490258898219, 204800000000000

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000583, A000584, A001014, A001017, A008454, A013669.

Cf. A004813 - A004823 (sums of 2, ..., 12 positive eleventh powers).

[n^11: n in [0..40]]; // Vincenzo Librandi, Jul 05 2014

Table[n^11, {n, 0, 30}] (* Vincenzo Librandi, Jul 05 2014 *)

A008455(n):=n^11$ makelist(A008455(n),n,0,20); /* Martin Ettl, Dec 17 2012 */

A008455(n)=n^11 \\ M. F. Hasler, Jul 03 2025

A008455 = lambda n: n**11 # M. F. Hasler, Jul 03 2025

a(n) = A000584(n)*A001014(n).
Multiplicative with a(p^e) = p^(11*e). - David W. Wilson, Aug 01 2001
Totally multiplicative with a(p) = p^11 for primes p. - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(11) (A013669).
Sum_{n>=1} (-1)^(n+1)/a(n) = 1023*zeta(11)/1024. (End)

A003368 Numbers that are the sum of 12 positive 6th powers.

Original entry on oeis.org

12, 75, 138, 201, 264, 327, 390, 453, 516, 579, 642, 705, 740, 768, 803, 866, 929, 992, 1055, 1118, 1181, 1244, 1307, 1370, 1433, 1468, 1531, 1594, 1657, 1720, 1783, 1846, 1909, 1972, 2035, 2098, 2196, 2259, 2322, 2385, 2448, 2511, 2574, 2637, 2700, 2763, 2924, 2987

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
54710 is in the sequence as 54710 = 2^6 + 3^6 + 3^6 + 3^6 + 3^6 + 4^6 + 4^6 + 4^6 + 4^6 + 4^6 + 5^6 + 5^6.
94302 is in the sequence as 94302 = 1^6 + 1^6 + 1^6 + 1^6 + 1^6 + 2^6 + 2^6 + 2^6 + 2^6 + 3^6 + 6^6 + 6^6.
133585 is in the sequence as 133585 = 1^6 + 1^6 + 1^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 4^6 + 4^6 + 4^6 + 7^6. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A001014 (sixth powers).

Cf. A003358 - A003367 (numbers that are the sum of 2, ..., 11 positive 6th powers); A003335, A003346, A003357, A003379, A003390, A004801, A004812, A004823 (numbers that are the sum of 12 positive 3rd, ..., 11th powers).

Module[{upto=2200,r},r=Ceiling[Surd[upto,6]];Select[Union[Total/@ Tuples[ Range[r]^6,12]],#<=upto&]] (* Harvey P. Dale, Aug 25 2015 *)

(A003368_upto(N, k=12, m=6)=[n|n<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, n)])(3000) \\ 2nd & 3rd optional arg allow to get other sequences of this group. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

A003362 Numbers that are the sum of 6 positive 6th powers.

Original entry on oeis.org

6, 69, 132, 195, 258, 321, 384, 734, 797, 860, 923, 986, 1049, 1462, 1525, 1588, 1651, 1714, 2190, 2253, 2316, 2379, 2918, 2981, 3044, 3646, 3709, 4101, 4164, 4227, 4290, 4353, 4374, 4416, 4829, 4892, 4955, 5018, 5081, 5557, 5620, 5683, 5746, 6285, 6348, 6411, 7013

Offset: 1

N. J. A. Sloane

This sequence has no sixth powers less than 1.5133*10^35. - J. Lowell, Jul 03 2021

			From _David A. Corneth_, Aug 03 2020: (Start)
448832 is in the sequence as 448832 = 2^6 + 6^6 + 6^6 + 6^6 + 6^6 + 8^6.
1000733 is in the sequence as 1000733 = 1^6 + 1^6 + 1^6 + 1^6 + 3^6 + 10^6.
1819677 is in the sequence as 1819677 = 1^6 + 1^6 + 3^6 + 3^6 + 6^6 + 11^6. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A001014 (sixth powers).

Column k=6 of A336725.

Reap[For[n = 1, n <= 10000, n++, If[AnyTrue[PowersRepresentations[n, 6, 6], First[#] > 0&], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 18 2017 *)
With[{nn=5},Select[Total/@Tuples[Range[nn]^6,6]//Union,#<=nn^6-5&]] (* Harvey P. Dale, Mar 27 2022 *)

A003360 Numbers that are the sum of 4 positive 6th powers.

Original entry on oeis.org

4, 67, 130, 193, 256, 732, 795, 858, 921, 1460, 1523, 1586, 2188, 2251, 2916, 4099, 4162, 4225, 4288, 4827, 4890, 4953, 5555, 5618, 6283, 8194, 8257, 8320, 8922, 8985, 9650, 12289, 12352, 13017, 15628, 15691, 15754, 15817, 16356, 16384, 16419, 16482, 17084, 17147, 17812

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
7875074 is in the sequence as 7875074 = 5^6 + 6^6 + 12^6 + 13^6.
12770418 is in the sequence as 12770418 = 7^6 + 8^6 + 10^6 + 15^6.
20763201 is in the sequence as 20763201 = 1^6 + 10^6 + 12^6 + 16^6. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A001014 (sixth powers).

Reap[For[n = 1, n <= 20000, n++, If[AnyTrue[PowersRepresentations[n, 4, 6], First[#] > 0&], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 18 2017 *)

A003361 Numbers that are the sum of 5 positive 6th powers.

Original entry on oeis.org

5, 68, 131, 194, 257, 320, 733, 796, 859, 922, 985, 1461, 1524, 1587, 1650, 2189, 2252, 2315, 2917, 2980, 3645, 4100, 4163, 4226, 4289, 4352, 4828, 4891, 4954, 5017, 5556, 5619, 5682, 6284, 6347, 7012, 8195, 8258, 8321, 8384, 8923, 8986, 9049, 9651, 9714, 10379, 12290

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
1071803 is in the sequence as 1071803 = 3^6 + 4^6 + 4^6 + 9^6 + 9^6.
3563572 is in the sequence as 3563572 = 3^6 + 4^6 + 5^6 + 11^6 + 11^6.
4862156 is in the sequence as 4862156 = 1^6 + 4^6 + 5^6 + 5^6 + 13^6. (End)

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001014 (sixth powers).

With[{upto=10000},Select[Union[Total/@Tuples[Range[Floor[Surd[upto-4,6]]]^6,5]],#<=upto&]] (* Harvey P. Dale, Apr 01 2016 *)

A003363 Numbers that are the sum of 7 positive 6th powers.

Original entry on oeis.org

7, 70, 133, 196, 259, 322, 385, 448, 735, 798, 861, 924, 987, 1050, 1113, 1463, 1526, 1589, 1652, 1715, 1778, 2191, 2254, 2317, 2380, 2443, 2919, 2982, 3045, 3108, 3647, 3710, 3773, 4102, 4165, 4228, 4291, 4354, 4375, 4417, 4438, 4480, 4830, 4893, 4956, 5019, 5082

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
267342 is in the sequence as 267342 = 1^6 + 2^6 + 3^6 + 5^6 + 5^6 + 7^6 + 7^6.
594482 is in the sequence as 594482 = 1^6 + 4^6 + 4^6 + 4^6 + 4^6 + 6^6 + 9^6.
696667 is in the sequence as 696667 = 2^6 + 2^6 + 2^6 + 3^6 + 6^6 + 7^6 + 9^6. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A001014 (sixth powers).

Reap[For[n = 1, n <= 5000, n++, If[AnyTrue[PowersRepresentations[n, 7, 6], First[#] > 0&], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 18 2017 *)

a(n) == 0 (mod 7). - Joerg Arndt, Nov 08 2022

A003365 Numbers that are the sum of 9 positive 6th powers.

Original entry on oeis.org

9, 72, 135, 198, 261, 324, 387, 450, 513, 576, 737, 800, 863, 926, 989, 1052, 1115, 1178, 1241, 1465, 1528, 1591, 1654, 1717, 1780, 1843, 1906, 2193, 2256, 2319, 2382, 2445, 2508, 2571, 2921, 2984, 3047, 3110, 3173, 3236, 3649, 3712, 3775, 3838, 3901, 4104, 4167, 4230

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
124882 is in the sequence as 124882 = 2^6 + 2^6 + 2^6 + 2^6 + 2^6 + 5^6 + 5^6 + 6^6 + 6^6.
188939 is in the sequence as 188939 = 2^6 + 2^6 + 3^6 + 3^6 + 3^6 + 6^6 + 6^6 + 6^6 + 6^6.
257236 is in the sequence as 257236 = 3^6 + 3^6 + 4^6 + 4^6 + 4^6 + 4^6 + 4^6 + 7^6 + 7^6. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A001014 (sixth powers).

Reap[For[n = 1, n <= 3000, n++, If[AnyTrue[PowersRepresentations[n, 9, 6], First[#] > 0&], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 18 2017 *)

cp's OEIS Frontend

A001016 Eighth powers: a(n) = n^8.

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A081119 Number of integral solutions to Mordell's equation y^2 = x^3 + n.

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A054504 Numbers n such that Mordell's equation y^2 = x^3 + n has no integral solutions.

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A008455 11th powers: a(n) = n^11.

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A003368 Numbers that are the sum of 12 positive 6th powers.

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A003362 Numbers that are the sum of 6 positive 6th powers.

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A003360 Numbers that are the sum of 4 positive 6th powers.

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