A001015 -id:A001015 - cp's OEIS frontend

A101095 Fourth difference of fifth powers (A000584).

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

A168635 a(n) = n^7*(n + 1)/2.

Original entry on oeis.org

0, 1, 192, 4374, 40960, 234375, 979776, 3294172, 9437184, 23914845, 55000000, 116923026, 232906752, 439239619, 790601280, 1366875000, 2281701376, 3693048057, 5816090304, 8938717390, 13440000000, 19811973951, 28685115712, 40857905364, 57330892800

Offset: 0

N. J. A. Sloane, Dec 11 2009

G. C. Greubel, 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).

Sequences of the form n^7*(n^k + 1)/2: A001015 (k=0), this sequence (k=1), A168636 (k=2), A168660 (k=3), A168661 (k=4), A168662 (k=5), A168663 (k=6), A168664 (k=7), A168665 (k=8), A168666 (k=9), A168667 (k=10).

[n^7*(n+1)/2: n in [0..25]]; // Vincenzo Librandi, Jul 29 2016

Table[n^7*(n + 1)/2, {n,0,25}] (* G. C. Greubel, Jul 28 2016 *)

a(n)=n^7*(n+1)/2 \\ Charles R Greathouse IV, Jul 29 2016

def A168635(n): return n^6*binomial(n+1,2)
print([A168635(n) for n in range(31)]) # G. C. Greubel, Mar 23 2025

From G. C. Greubel, Jul 28 2016: (Start)
G.f.: x*(1 + 183*x + 2682*x^2 + 8422*x^3 + 7197*x^4 + 1611*x^5 + 64*x^6)/(1 - x)^9.
E.g.f.: (1/2)*x*(2 + 190*x + 1267*x^2 + 2051*x^3 + 1190*x^4 + 287*x^5 + 29*x^6 + x^7)*exp(x). (End)

A266314 Least positive integer x such that n + x^7 = y^2 + z^2 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

2, 1, 2, 13, 1, 3, 7, 1, 2, 1, 3, 15, 1, 6, 11, 11, 1, 1, 2, 1, 2, 2, 7, 3, 1, 1, 3, 5, 1, 2, 7, 1, 2, 1, 2, 5, 1, 4, 3, 1, 1, 2, 2, 7, 1, 2, 7, 3, 5, 1, 2, 1, 1, 2, 11, 21, 5, 1, 3, 5, 1, 3, 3, 3, 1, 2, 2, 1, 4, 2, 3

Offset: 0

Zhi-Wei Sun, Dec 27 2015

nonn

The general conjecture in A266277 implies that for each odd prime p and any integer m there are positive integers x, y and z such that m + x^p = y^2 + z^2.
For k = 4,6,8,... and any integer m == 6 (mod 8), there are no integers x, y and z with m + x^k = y^2 + z^2 since m + x^k with x an integer is congruent to 6 or 7 modulo 8.
As 2j+1 = (j+1)^2 - j^2, if m - z^k is odd with |m - z^k| > 1 then m + x^2 = y^2 + z^k for some positive integers x and y.

			a(2) = 2 since 2 + 2^7 = 3^2 + 11^2.
a(3) = 13 since 3 + 13^7 = 554^2 + 7902^2.
a(5) = 3 since 5 + 3^7 = 16^2 + 44^2.
a(6) = 7 since 6 + 7^7 = 30^2 + 907^2.
a(462) = 71 since 462 + 71^7 = 456497^2 + 2981062^2.

Zhi-Wei Sun and Chai Wah Wu, Table of n, a(n) for n = 0..10000 n = 0..2000 from Zhi-Wei Sun
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120. (Cf. Section 5.)

Cf. A000290, A001015, A266152, A266153, A266230, A266231, A266277.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[x=1;Label[bb];Do[If[SQ[n+x^7-y^2],Print[n," ",x];Goto[aa]],{y,1,Sqrt[(n+x^7)/2]}];x=x+1;Goto[bb];Label[aa];Continue,{n,1,70}]
(* second program: *)
xmax = 100; r[n_, x_] := Reduce[y>0 && z>0 && n+x^7 == y^2+z^2, {y, z}, Integers]; a[n_] := For[x=1, x <= xmax, x++, If[r[n, x] =!= False, Return[x]]] /. Null -> 0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 27 2015 *)

A016951 a(n) = (6*n + 3)^7.

Original entry on oeis.org

2187, 4782969, 170859375, 1801088541, 10460353203, 42618442977, 137231006679, 373669453125, 897410677851, 1954897493193, 3938980639167, 7446353252589, 13348388671875, 22876792454961, 37725479487783, 60170087060757, 93206534790699, 140710042265625, 207616015289871

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A016759, A016945, A016946, A016947, A016948, A016949, A016950.

Subsequence of A001015.

[(6*n+3)^7: n in [0..40]]; // Vincenzo Librandi, May 05 2011

a[n_] := (6*n + 3)^7; Array[a, 50, 0] (* Amiram Eldar, Mar 30 2022 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^7.
a(n) = 3^7*A016759(n).
Sum_{n>=0} 1/a(n) = 127*zeta(7)/279936.
Sum_{n>=0} (-1)^n/a(n) = 61*Pi^7/403107840. (End)

A016963 a(n) = (6*n + 4)^7.

Original entry on oeis.org

16384, 10000000, 268435456, 2494357888, 13492928512, 52523350144, 163840000000, 435817657216, 1028071702528, 2207984167552, 4398046511104, 8235430000000, 14645194571776, 24928547056768, 40867559636992, 64847759419264, 100000000000000, 150363025899136, 221068140740608

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 (8,-28,56,-70,56,-28,8,-1).

Cf. A016795, A016957, A016958, A016959, A016960, A016961, A016962.

Subsequence of A001015.

[(6*n+4)^7: n in [0..20]]; // Vincenzo Librandi, May 07 2011

(6*Range[0,20]+4)^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{16384,10000000,268435456,2494357888,13492928512,52523350144,163840000000,435817657216},20] (* Harvey P. Dale, Mar 03 2018 *)

From Amiram Eldar, Mar 31 2022: (Start)
a(n) = A016957(n)^7.
a(n) = 2^7*A016795(n).
Sum_{n>=0} 1/a(n) = 1093*zeta(7)/279936 - 7*Pi^7/(3149280*sqrt(3)). (End)

A016975 a(n) = (6*n + 5)^7.

Original entry on oeis.org

78125, 19487171, 410338673, 3404825447, 17249876309, 64339296875, 194754273881, 506623120463, 1174711139837, 2488651484819, 4902227890625, 9095120158391, 16048523266853, 27136050989627, 44231334895529, 69833729609375, 107213535210701, 160578147647843, 235260548044817

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 (8,-28,56,-70,56,-28,8,-1).

Cf. A016969 (6*n + 5), A016970, A016971, A016972, A016973, A016974.

Subsequence of A001015 (n^7).

[(6*n+5)^7: n in [0..25]]; // Vincenzo Librandi, May 11 2011

(6Range[0,20]+5)^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{78125,19487171,410338673,3404825447,17249876309,64339296875,194754273881,506623120463},20] (* Harvey P. Dale, Jan 30 2013 *)

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). - Harvey P. Dale, Jan 30 2013
From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016969(n)^7.
Sum_{n>=0} 1/a(n) = 138811*zeta(7)/279936 - 301*Pi^7/(1049760*sqrt(3)). (End)

A113852 Numbers whose prime factors are raised to the seventh power.

Original entry on oeis.org

128, 2187, 78125, 279936, 823543, 10000000, 19487171, 62748517, 105413504, 170859375, 410338673, 893871739, 1801088541, 2494357888, 3404825447, 8031810176, 17249876309, 21870000000, 27512614111, 42618442977, 52523350144, 64339296875, 94931877133, 114415582592

Offset: 1

Cino Hilliard, Jan 25 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000

Proper subset of A001015.
Nonunit terms of A329332 column 7 in ascending order.
Cf. A005117, A013665, A013672.

Select[Range@34^7, Union[Last /@ FactorInteger@# ] == {7} &] (* Robert G. Wilson v, Jan 26 2006 *)
Select[Range[2, 34], SquareFreeQ]^7 (* Amiram Eldar, Oct 13 2020 *)

allpwrfact(n,p) = /* All prime factors are raised to the power p */ { local(x,j,ln,y,flag); for(x=4,n, y=Vec(factor(x)); ln = length(y[1]); flag=0; for(j=1,ln, if(y[2][j]==p,flag++); ); if(flag==ln,print1(x",")); ) }

from math import isqrt
from sympy import mobius
def A113852(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**7 # Chai Wah Wu, Feb 25 2025

From Amiram Eldar, Oct 13 2020: (Start)
a(n) = A005117(n+1)^7.
Sum_{n>=1} 1/a(n) = zeta(7)/zeta(14) - 1. (End)

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

A343287 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^(k^7).

Original entry on oeis.org

1, 128, 2187, 24640, 78125, 559872, 823543, 4552064, 7175547, 20000000, 19487171, 125551296, 62748517, 210827008, 341718750, 818079776, 410338673, 2142910080, 893871739, 4485000000, 3602177082, 4988715776, 3404825447, 26025929856, 9155312500, 16063620352, 22666490820

Offset: 1

Ilya Gutkovskiy, Apr 10 2021

nonn

Cf. A001015, A001055, A023876, A050367, A329125, A343283, A343284, A343285, A343286, A343288, A343289.

A010802 14th powers: a(n) = n^14.

Original entry on oeis.org

0, 1, 16384, 4782969, 268435456, 6103515625, 78364164096, 678223072849, 4398046511104, 22876792454961, 100000000000000, 379749833583241, 1283918464548864, 3937376385699289, 11112006825558016, 29192926025390625, 72057594037927936, 168377826559400929, 374813367582081024

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 (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A013672 (zeta(14)), A001015 (n^7).

Cf. A000290, (squares), A000578, (cubes), A000583, (4th powers), A000584, (5th powers), A008455 (11th powers).

[n^14: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

Range[0,20]^14 (* Harvey P. Dale, Nov 08 2011 *)

for(n=0,15,print1(n^14,", ")) \\ Derek Orr, Feb 27 2017

A010802(n)=n^14 \\ M. F. Hasler, Jul 03 2025

A010802 = lambda n: n**14 # M. F. Hasler, Jul 03 2025

Totally multiplicative with a(p) = p^14 for prime p. Multiplicative with a(p^e) = p^(14e). - Jaroslav Krizek, Nov 01 2009
From Ilya Gutkovskiy, Feb 27 2017: (Start)
Dirichlet g.f.: zeta(s-14).
Sum_{n>=1} 1/a(n) = 2*Pi^14/18243225 = A013672. (End)
a(n) = A001015(n)^2. - Michel Marcus, Feb 28 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = 8191*zeta(14)/8192 = 8191*Pi^14/74724249600. - Amiram Eldar, Oct 08 2020

A132214 Numbers that are sums of seventh powers of two distinct primes.

Original entry on oeis.org

2315, 78253, 80312, 823671, 825730, 901668, 19487299, 19489358, 19565296, 20310714, 62748645, 62750704, 62826642, 63572060, 82235688, 410338801, 410340860, 410416798, 411162216, 429825844, 473087190, 893871867, 893873926

Offset: 1

Jonathan Vos Post, Aug 13 2007

This is to 7th powers as A130555 is to 6th powers, A130292 is to fifth powers, A130873 is to 4th powers and A120398 is to cubes. These can never be prime, as the polynomial x^7 + y^7 factors over Z. Note however that A132215, which is the analog for eighth powers, can be prime, as seen also in A132216.

			a(1) = 2^7 + 3^7 = 2315 = 5 * 463.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001015, A050997, A120398, A122616, A130873, A130555, A132215, A132216.

P:= select(isprime, [2,seq(i,i=3..100,2)]): nP:= nops(P):
N:= 2^7 + P[-1]^7:
sort(convert(select(`<=`, {seq(seq(P[i]^7+P[j]^7,j=i+1..nP),i=1..nP-1)},N),list)); # Robert Israel, Jul 01 2024

Select[Sort[ Flatten[Table[Prime[n]^7 + Prime[k]^7, {n, 15}, {k, n - 1}]]], # <= Prime[15^7] &]
Union[Total/@(Subsets[Prime[Range[10]],{2}]^7)] (* Harvey P. Dale, Jan 03 2012 *)

{A001015(A000040(i)) + A001015(A000040(j)) for i > j}.

cp's OEIS Frontend

A101095 Fourth difference of fifth powers (A000584).

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A168635 a(n) = n^7*(n + 1)/2.

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A266314 Least positive integer x such that n + x^7 = y^2 + z^2 for some positive integers y and z, or 0 if no such x exists.

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A016951 a(n) = (6*n + 3)^7.

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A016963 a(n) = (6*n + 4)^7.

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Magma

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A016975 a(n) = (6*n + 5)^7.

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A113852 Numbers whose prime factors are raised to the seventh power.

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A343287 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^(k^7).