A001014 -id:A001014 - cp's OEIS frontend

A016974 a(n) = (6*n + 5)^6.

15625, 1771561, 24137569, 148035889, 594823321, 1838265625, 4750104241, 10779215329, 22164361129, 42180533641, 75418890625, 128100283921, 208422380089, 326940373369, 496981290961, 735091890625, 1061520150601, 1500730351849, 2081951752609, 2839760855281, 3814697265625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A016969 (6*n +5), A016970, A016971, A016972, A016973.

Subsequence of A001014 (n^6).

[(6*n+5)^6: n in [0..25]]; // Vincenzo Librandi, May 10 2011

(6Range[0,20]+5)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{15625,1771561,24137569,148035889,594823321,1838265625,4750104241},30] (* Harvey P. Dale, Apr 24 2025 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^6 = A016970(n)^3 = A016971(n)^2.
Sum_{n>=0} 1/a(n) = PolyGamma(5, 5/6)/5598720. (End)

A101095 Fourth difference of fifth powers (A000584).

Original entry on oeis.org

1, 28, 121, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, 2040, 2160, 2280, 2400, 2520, 2640, 2760, 2880, 3000, 3120, 3240, 3360, 3480, 3600, 3720, 3840, 3960, 4080, 4200, 4320, 4440, 4560, 4680, 4800, 4920, 5040, 5160, 5280

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original Name: Shells (nexus numbers) of shells of shells of shells of the power of 5.

The (Worpitzky/Euler/Pascal Cube) "MagicNKZ" algorithm is: MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n, with k>=0, n>=1, z>=0. MagicNKZ is used to generate the n-th accumulation sequence of the z-th row of the Euler Triangle (A008292). For example, MagicNKZ(3,k,0) is the 3rd row of the Euler Triangle (followed by zeros) and MagicNKZ(10,k,1) is the partial sums of the 10th row of the Euler Triangle. This sequence is MagicNKZ(5,k-1,2).

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Archive Machine link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883
Eric Weisstein, Link to section of MathWorld: Eulerian Number
Eric Weisstein, Link to section of MathWorld: Nexus number
Eric Weisstein, Link to section of MathWorld: Finite Differences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Fourth differences of A000584, third differences of A022521, second differences of A101098, and first differences of A101096.
For other sequences based upon MagicNKZ(n,k,z):
...... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7  |  n = 8
--------------------------------------------------------------------------------------
z =  0 | A000007 | A019590 | .......  MagicNKZ(n,k,0) = T(n,k+1) from A008292  .......
z =  1 | A000012 | A040000 | A101101 | A101104 | A101100 | ....... | ....... | .......
z =  2 | A000027 | A005408 | A008458 | A101103 | thisSeq | ....... | ....... | .......
z =  3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | ....... | .......
z =  4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | ....... | .......
z =  5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181 | .......
z =  6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177 | A255182
z =  7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523 | A255178
z =  8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015 | A022524
z =  9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541 | A001016
z = 10 | A000582 | A054333 | A254469 | A254681 | A254644 | A254640 | A250212 | A000542
z = 11 | A001287 | A054334 | A254869 | A254470 | A254682 | A254645 | A254641 | A253636
z = 12 | A001288 | A057788 | ....... | A254870 | A254471 | A254683 | A254646 | A254642
z = 13 | A010965 | ....... | ....... | ....... | A254871 | A254472 | A254684 | A254647
z = 14 | A010966 | ....... | ....... | ....... | ....... | A254872 | ....... | .......
--------------------------------------------------------------------------------------
Cf. A047969.

I:=[1,28,121,240,360]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, May 07 2015

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 5, 5}, {z, 2, 2}, {k, 0, 34}]
CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)
Join[{1,28,121,240},Differences[Range[50]^5,4]] (* or *) LinearRecurrence[{2,-1},{1,28,121,240,360},50] (* Harvey P. Dale, Jun 11 2016 *)

a(n)=if(n>3, 120*n-240, 33*n^2-72*n+40) \\ Charles R Greathouse IV, Oct 11 2015

[1,28,121]+[120*(k-2) for k in range(4,36)] # Danny Rorabaugh, Apr 23 2015

a(k+1) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n; n = 5, z = 2.
For k>3, a(k) = Sum_{j=0..4} (-1)^j*binomial(4, j)*(k - j)^5 = 120*(k - 2).
a(n) = 2*a(n-1) - a(n-2), n>5. G.f.: x*(1+26*x+66*x^2+26*x^3+x^4) / (1-x)^2. - Colin Barker, Mar 01 2012

MagicNKZ material edited, Crossrefs table added, SeriesAtLevelR material removed by Danny Rorabaugh, Apr 23 2015

Name changed and keyword 'uned' removed by Danny Rorabaugh, May 06 2015

A106318 Bhaskara twins: n such that 2n^2 = X^3 and 2n^3 = Y^2.

Original entry on oeis.org

2, 128, 1458, 8192, 31250, 93312, 235298, 524288, 1062882, 2000000, 3543122, 5971968, 9653618, 15059072, 22781250, 33554432, 48275138, 68024448, 94091762, 128000000, 171532242, 226759808, 296071778, 382205952, 488281250, 617831552

Offset: 1

Lekraj Beedassy, Apr 29 2005

S. S. Gupta, 'Bhaskara Pairs' in 'Science Today' (subsequently renamed '2001'), January 1988, pp. 68, Times of India, Mumbai.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Richard J. Mathar, Construction of Bhaskara Pairs, arXiv:1703.01677 [math.NT], 2017.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A106319, A106320, A106321, A106322.

Cf. A001014.

a106318 = (* 2) . (^ 6)  -- Reinhard Zumkeller, May 27 2015

LinearRecurrence[{7,-21,35,-35,21,-7,1},{2,128,1458,8192,31250,93312,235298},30] (* Harvey P. Dale, May 11 2017 *)

a(n)=2*n^6 \\ Charles R Greathouse IV, Feb 09 2012

a(n) = 2*n^6 = 2*A001014(n).
G.f.: 2*(1+x)*(1+56*x+246*x^2+56*x^3+x^4)/(1-x)^7. - Colin Barker, Apr 18 2012
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Wesley Ivan Hurt, Apr 23 2021

A113851 Numbers whose prime factors are raised to the sixth power.

Original entry on oeis.org

64, 729, 15625, 46656, 117649, 1000000, 1771561, 4826809, 7529536, 11390625, 24137569, 47045881, 85766121, 113379904, 148035889, 308915776, 594823321, 729000000, 887503681, 1291467969, 1544804416, 1838265625, 2565726409, 3010936384, 3518743761, 4750104241

Offset: 1

Cino Hilliard, Jan 25 2006

Nathaniel Johnston, Table of n, a(n) for n = 1..5000

Subset of A001014. Superset of A030516.
Nonunit terms of A329332 column 6 in ascending order.
Cf. A005117, A269404.

for n from 2 to 100 do if(numtheory[issqrfree](n))then printf("%d, ", n^6): fi: od: # Nathaniel Johnston, Jun 21 2011

Select[ Range@37^6, Union[Last /@ FactorInteger@# ] == {6} &] (* Robert G. Wilson v *)
Select[Range[2, 37], SquareFreeQ]^6 (* Amiram Eldar, Oct 13 2020 *)

from math import isqrt
from sympy import mobius
def A113851(n):
    def f(x): return int(n+1-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**6 # Chai Wah Wu, Feb 25 2025

a(n) = A005117(n+1)^6. - Nathaniel Johnston, Jun 21 2011

Sum_{n>=1} 1/a(n) = zeta(6)/zeta(12) - 1 = A269404 - 1. - Amiram Eldar, Oct 13 2020

More terms from Robert G. Wilson v, Jan 26 2006

A271099 Number of ordered ways to write n as u^3 + v^3 + 2x^3 + 2y^3 + 3*z^3, where u, v, x, y and z are nonnegative integers with u <= v and x <= y.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 3, 1, 3, 3, 3, 3, 1, 2, 2, 2, 3, 4, 4, 3, 4, 2, 5, 3, 4, 5, 2, 4, 1, 1, 4, 2, 4, 3, 4, 1, 2, 1, 3, 2, 1, 4, 1, 2, 4, 2, 7, 4, 5, 5, 2, 3, 2, 3, 3, 4, 2, 5, 4, 3, 6

Offset: 0

Zhi-Wei Sun, Mar 30 2016

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 10, 14, 15, 17, 22, 38, 39, 45, 47, 50, 52, 76, 102, 103, 188, 295, 366, 534.
(ii) Any natural number n can be written as s^4 + t^4 + 2*u^4 + 2*v^4 + 3*x^4 + 3*y^4 + 7*z^4, where s, t, u, v, x, y and z are nonnegative integers. Also, each natural number n can be written as r^5 + s^5 +  t^5 + u^5 + 2*v^5 + 4*w^5 + 6*x^5 + 9*y^5 +12*z^5, where r, s, t, u, v, w, x, y and z are nonnegative integers.
(iii) In general, for any integer k > 2, there are 2*k-1 positive integers c(1), c(2), ..., c(2k-1) such that {c(1)*x(1)^k + c(2)*x(2)^k + ... + c(2k-1)*x(2k-1)^k: x(1),x(2),...,x(k) = 0,1,2,...} = {0,1,2,3,...} and that c(1)+c(2)+...+c(2k-1) = g(k), where g(k) = 2^k+floor((3/2)^k)-2 as given by A002804.
This conjecture is stronger than the classical Waring problem on sums of k-th powers. Concerning parts (i) and (ii) of the conjecture, we note that 1+1+2+2+3 = 9 = g(3), 1+1+2+2+3+3+7 = 19 = g(4) and 1+1+1+1+2+4+6+9+12 = 37 = g(5).
We have verified that a(n) > 0 for all n = 0..10^6, and that part (ii) of the conjecture holds for n up to 10^5. Concerning part (iii) for k = 6, we conjecture that any natural number can be written as x(1)^6+x(2)^6+x(3)^6+x(4)^6+x(5)^6+3*x(6)^6+5*x(7)^6+6*x(8)^6+10*x(9)^6+18*x(10)^6+26*x(11)^6 with x(1),x(2),...,x(11) nonnegative integers. Note that 1+1+1+1+1+3+5+6+10+18+26 = 73 = g(6). - Zhi-Wei Sun, Mar 31 2016

			a(1) = 1 since 1 = 0^3 + 1^3 + 2*0^3 + 2*0^3 + 3*0^3.
a(10) = 1 since 10 = 0^3 + 2^3 + 2*0^3 + 2*1^3 + 3*0^3.
a(14) = 1 since 14 = 1^3 + 2^3 + 2*0^3 + 2*1^3 + 3*1^3.
a(15) = 1 since 15 = 0^3 + 2^3 + 2*1^3 + 2*1^3 + 3*1^3.
a(17) = 1 since 17 = 0^3 + 1^3 + 2*0^3 + 2*2^3 + 3*0^3.
a(22) = 1 since 22 = 0^3 + 1^3 + 2*1^3 + 2*2^3 + 3*1^3.
a(38) = 1 since 38 = 2^3 + 3^3 + 2*0^3 + 2*0^3 + 3*1^3.
a(39) = 1 since 39 = 2^3 + 3^3 + 2*1^3 + 2*1^3 + 3*0^3.
a(45) = 1 since 45 = 0^3 + 3^3 + 2*1^3 + 2*2^3 + 3*0^3.
a(47) = 1 since 47 = 1^3 + 3^3 + 2*0^3 + 2*2^3 + 3*1^3.
a(50) = 1 since 50 = 0^3 + 2^3 + 2*1^3 + 2*2^3 + 3*2^3.
a(52) = 1 since 52 = 1^3 + 3^3 + 2*0^3 + 2*0^3 + 3*2^3.
a(76) = 1 since 76 = 2^3 + 4^3 + 2*1^3 +2*1^3 + 3*0^3.
a(102) = 1 since 102 = 0^3 + 2^3 + 2*2^3 + 2*3^3 + 3*2^3.
a(103) = 1 since 103 = 1^3 + 2^3 + 2*2^3 + 2*3^3 + 3*2^3.
a(188) = 1 since 188 = 3^3 + 4^3 + 2*0^3 + 2*2^3 + 3*3^3.
a(295) = 1 since 295 = 1^3 + 6^3 + 2*0^3 + 2*3^3 + 3*2^3.
a(366) = 1 since 366 = 2^3 + 3^3 + 2*0^3 + 2*5^3 + 3*3^3.
a(534) = 1 since 534 = 1^3 + 8^3 + 2*1^3 + 2*2^3 + 3*1^3.

M. B. Nathanson, Additive Number Theory: The Classical Bases, Grad. Texts in Math., Vol 164, Springer, 1996, Chapters 2 and 3.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000578, A000583, A000584, A001014, A002804.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[r=0;Do[If[CQ[n-3z^3-2x^3-2y^3-u^3],r=r+1],{z,0,(n/3)^(1/3)},{x,0,((n-3z^3)/4)^(1/3)},{y,x,((n-3z^3-2x^3)/2)^(1/3)},{u,0,((n-3z^3-2x^3-2y^3)/2)^(1/3)}];Print[n," ",r];Continue,{n,0,70}]

A001120 a(0) = a(1) = 1; for n > 1, a(n) = n*a(n-1) + (-1)^n.

Original entry on oeis.org

1, 1, 3, 8, 33, 164, 985, 6894, 55153, 496376, 4963761, 54601370, 655216441, 8517813732, 119249392249, 1788740883734, 28619854139745, 486537520375664, 8757675366761953, 166395831968477106, 3327916639369542121, 69886249426760384540, 1537497487388728459881

Offset: 0

N. J. A. Sloane, Simon Plouffe

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).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 1..100 from T. D. Noe)
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]

A000166 has a similar recurrence.
Column k=1 of A334715.
Cf. A000142.

a:= proc(n) option remember;
     `if`(n<2, 1, n*a(n-1)+(-1)^n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 09 2020

t = {1}; Do[AppendTo[t, n*t[[-1]] + (-1)^n], {n, 2, 20}]; t (* T. D. Noe, Jun 26 2012 *)
RecurrenceTable[{a[1]==1,a[n]==n a[n-1]+(-1)^n},a,{n,30}] (* Harvey P. Dale, Feb 19 2018 *)

PARI
```
a(n)=if(n<2,n>0,n*a(n-1)+(-1)^n)
```

a(n)=if(n<1,0,n!*polcoeff((1+exp(-x+x*O(x^n)))/(1-x),n))

Nearest integer to n!(1+1/e).
a(n) = A000166(n) + [n > 0] * n!. - Philippe Deléham, Sep 04 2005
a(n) = (n-1)*(a(n-1)+a(n-2)), n>2. - Gary Detlefs, Apr 11 2010
E.g.f.: (exp(x)*x+1)*exp(-x)/(1-x). - Alois P. Heinz, May 08 2020

a(0)=1 prepended by Alois P. Heinz, May 08 2020

A002604 a(n) = n^6 + 1.

Original entry on oeis.org

1, 2, 65, 730, 4097, 15626, 46657, 117650, 262145, 531442, 1000001, 1771562, 2985985, 4826810, 7529537, 11390626, 16777217, 24137570, 34012225, 47045882, 64000001, 85766122, 113379905, 148035890

Offset: 0

N. J. A. Sloane

Because of Fermat's little theorem, a(n) is never divisible by 7. - Altug Alkan, Apr 08 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Equals A001014 + 1. Cf. A024004, A002522.

[n^6 + 1: n in [0..50]]; // Vincenzo Librandi, May 02 2011

Table[n^6+1,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,2,65,730,4097,15626,46657},30] (* Harvey P. Dale, Jul 28 2021 *)

PARI
```
a(n)=n^6+1
```

G.f. (-1 + 5*x - 72*x^2 - 282*x^3 - 317*x^4 - 51*x^5 - 2*x^6) / (x - 1)^7. - R. J. Mathar, Aug 06 2012
Sum_{n>=0} 1/a(n) = 1/2 + Pi * (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi))) / 6 = 1.5171007340332164261529... . - Vaclav Kotesovec, Feb 14 2015
Sum_{n>=0} (-1)^n/a(n) = 1/2 + Pi/(6*sinh(Pi)) + Pi * (sqrt(3)*cosh(Pi/2) * sin((sqrt(3)*Pi)/2) + cos((sqrt(3)*Pi)/2) * sinh(Pi/2)) / (3*(cosh(Pi) - cos(sqrt(3)*Pi))) = 0.514210347292695053493... . - Vaclav Kotesovec, Feb 14 2015

A080761 Positive numbers of the form y^2 - x^3, x and y >= 1.

Original entry on oeis.org

1, 3, 8, 9, 12, 15, 17, 18, 19, 22, 24, 28, 30, 35, 36, 37, 38, 40, 41, 44, 48, 54, 55, 56, 57, 63, 64, 65, 68, 71, 73, 79, 80, 89, 92, 94, 97, 98, 99, 100, 101, 105, 106, 107, 108, 112, 113, 117, 119, 120, 121, 128, 129, 131, 132, 136, 138, 141, 142, 143, 145, 148, 151

Offset: 1

Cino Hilliard, Mar 10 2003

nonn

From Artur Jasinski, Oct 03 2007: (Start)
Some numbers have multiple partitions:
  8 = 4^2 - 8^3 = 312^2 - 46^3,
  9 = 6^2 - 3^3 = 15^2 - 6 ^3 = 253^2 - 40^3. (End)
This is Mordell's equation with the condition that x and y are positive. Sequence A054504 lists the n for which there is no solution to Mordell's equation. Hence, none of those numbers will be in this sequence. The terms of this sequence can be determined by looking at the link to Gebel's data. - T. D. Noe, Mar 23 2011

			8 is in the sequence since 3^2 = 1^3 + 8.

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]
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.
Cino Hilliard, Proof that n^3+7 <> k^2 for all integers n,k.

Complement of A080762.
Cf. sequences for n^3+7, n^3+17, n^3+3, n^3+2, n^3+5.
Cf. A029727, A029728, A134042, A134043.

With[{nn=100},Take[Union[Select[First[#]^2-Last[#]^3&/@Tuples[Range[ 20nn],2],#>0&]],nn]] (* Harvey P. Dale, Jul 10 2012 *)

diop(n,m) = { for(p=1,m, for(x=1,n, y=x*x*x+p; if(issquare(y),print1(p" "); break) ) ) }

"Positive" added to definition by N. J. A. Sloane, Oct 06 2007

A085539 a(n) = n^6 - n^5.

Original entry on oeis.org

0, 0, 32, 486, 3072, 12500, 38880, 100842, 229376, 472392, 900000, 1610510, 2737152, 4455516, 6991712, 10631250, 15728640, 22717712, 32122656, 44569782, 60800000, 81682020, 108226272, 141599546, 183140352, 234375000, 297034400, 373071582, 464679936

Offset: 0

N. J. A. Sloane, Jul 05 2003

For n>=1, a(n) is equal to the number of functions f:{1,2,3,4,5,6}->{1,2,...,n} such that for a fixed x in {1,2,3,4,5,6} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

A diagonal of A228273.

Cf. A000584, A001014.

[n^6-n^5: n in [0..40]]; // Vincenzo Librandi, Aug 15 2016

f[n_]:=n^6-n^5;f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
Table[n^6 - n^5, {n, 0, 50}] (* Vincenzo Librandi, Aug 15 2016 *)

a(n)=n^6-n^5 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: -2*x^2*(x^4+41*x^3+171*x^2+131*x+16)/(x-1)^7. - Colin Barker, Nov 06 2012

Sum_{n>=2} 1/a(n) = 5 - Sum_{k=2..5} zeta(k). - Amiram Eldar, Jul 05 2020

A163285 Triangle read by rows in which row n lists n+1 terms, starting with n^5 and ending with n^6, such that the difference between successive terms is equal to n^5 - n^4.

Original entry on oeis.org

0, 1, 1, 32, 48, 64, 243, 405, 567, 729, 1024, 1792, 2560, 3328, 4096, 3125, 5625, 8125, 10625, 13125, 15625, 7776, 14256, 20736, 27216, 33696, 40176, 46656, 16807, 31213, 45619, 60025, 74431, 88837, 103243, 117649, 32768, 61440, 90112, 118784, 147456

Offset: 0

Omar E. Pol, Jul 24 2009

The first term of row n is A000584(n) and the last term of row n is A001014(n).
The main entry for this sequence is A159797. See also A163282, A163283 and A163284.
Row sums give A163275. - Omar E. Pol, Mar 18 2012

			Triangle begins:
0;
1,1;
32,48,64;
243,405,567,729;
1024,1792,2560,3328,4096;
3125,5625,8125,10625,13125,15625;
7776,14256,20736,27216,33696,40176,46656;
16807,31213,45619,60025,74431,88837,103243,117649;
32768,61440,90112,118784,147456,176128,204800,233472,262144;
59049,111537,164025,216513,269001,321489,373977,426465,478953,531441;
100000,190000,280000,370000,460000,550000,640000,730000,820000,910000,1000000;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000584, A001014, A085538, A159797, A162611, A162614, A162622, A163282, A163283, A163284.

rw[n_]:=Range[n^5,n^6,n^5-n^4]; Join[{0,1},Flatten[Array[rw,10]]] (* Harvey P. Dale, Mar 18 2012 *)

A163285(n, k)=n^5 +k*(n^5 -n^4) \\ G. C. Greubel, Dec 17 2016

cp's OEIS Frontend

A016974 a(n) = (6*n + 5)^6.

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A101095 Fourth difference of fifth powers (A000584).

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A106318 Bhaskara twins: n such that 2*n^2 = X^3 and 2*n^3 = Y^2.

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A113851 Numbers whose prime factors are raised to the sixth power.

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A271099 Number of ordered ways to write n as u^3 + v^3 + 2*x^3 + 2*y^3 + 3*z^3, where u, v, x, y and z are nonnegative integers with u <= v and x <= y.

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A001120 a(0) = a(1) = 1; for n > 1, a(n) = n*a(n-1) + (-1)^n.

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A002604 a(n) = n^6 + 1.

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A106318 Bhaskara twins: n such that 2n^2 = X^3 and 2n^3 = Y^2.

A271099 Number of ordered ways to write n as u^3 + v^3 + 2x^3 + 2y^3 + 3*z^3, where u, v, x, y and z are nonnegative integers with u <= v and x <= y.