A010801 -id:A010801 - cp's OEIS frontend

A155016 Integer part of square root of A010801.

0, 1, 90, 1262, 8192, 34938, 114283, 311269, 741455, 1594323, 3162277, 5875603, 10343751, 17403307, 28172943, 44115700, 67108864, 99521746, 144301645, 205068240, 286216701, 393029741, 531798888, 709955183, 936209559, 1220703125

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 19 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 100 terms from Karl V. Keller, Jr.)

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), A155013 (k=5), A155014 (k=7), A155015 (k=11), this sequence (k=13), A155018 (k=15), A155019 (k=17).

[Floor(Sqrt(n^13)): n in [1..30]]; // G. C. Greubel, Dec 30 2017

a={};Do[AppendTo[a,IntegerPart[(n^13)^(1/2)]],{n,0,5!}];a
Table[Floor[Sqrt[n^13]], {n,1,30}] (* G. C. Greubel, Dec 30 2017 *)

for(n=1,30, print1(floor(sqrt(n^13)), ", ")) \\ G. C. Greubel, Dec 30 2017

Offset corrected by Karl V. Keller, Jr., Sep 27 2014

A253713 Second partial sums of 13th powers (A010801).

Original entry on oeis.org

1, 8194, 1610710, 70322090, 1359736595, 15709845116, 126948964044, 787943896860, 3990804658005, 17193665419150, 64919238324226, 219638016608374, 677231901484775, 1928540559615320, 5126044286105240, 12827147639965656, 30432829026732009, 68861475279169530, 149343104993864110, 311744734708558690, 628618742162372731

Offset: 1

Luciano Ancora, Jan 12 2015

The formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) are the m-th Faulhaber's formulas.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Luciano Ancora, Recurrence relation for the second partial sums of m-th powers
Luciano Ancora, Second partial sums of the m-th powers

[n*(n+1)*(n+2)*(6*n^12+72*n^11+297*n^10+330*n^9-765*n^8-1368*n^7+2059*n^6+2994*n^5-4091*n^4-2724*n^3+4069*n^2+66*n-735)/1260: n in [1..30]]; // Vincenzo Librandi, Jan 19 2015

Table[n (n+1) (n+2) (6 n^12 + 72 n^11 + 297 n^10 + 330 n^9 - 765 n^8 - 1368 n^7 + 2059 n^6 + 2994 n^5 - 4091 n^4 -2724 n^3 + 4069 n^2 + 66 n - 735) / 1260, {n, 40}] (* Vincenzo Librandi, Jan 19 2015 *)
Nest[Accumulate,Range[30]^13,2] (* Harvey P. Dale, Jul 24 2018 *)

a(n) = n*(n+1)*(n+2)*(6*n^12+72*n^11+297*n^10+330*n^9-765*n^8-1368*n^7+2059*n^6+2994*n^5-4091*n^4-2724*n^3+4069*n^2+66*n-735)/1260.

a(n) = 2*a(n-1)-a(n-2)+n^13.

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

Original entry on oeis.org

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

A008456 12th powers: a(n) = n^12.

Original entry on oeis.org

0, 1, 4096, 531441, 16777216, 244140625, 2176782336, 13841287201, 68719476736, 282429536481, 1000000000000, 3138428376721, 8916100448256, 23298085122481, 56693912375296, 129746337890625, 281474976710656, 582622237229761

Offset: 0

N. J. A. Sloane

Numbers which are square, cubic and quartic. - Doug Bell, Jun 03 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

a(n) = A123868(n) + 1.
Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A001014 (6th powers), A008454 (10th powers), A008455 (11th powers), A010801 (13th powers).
Cf. A013670 (zeta(12)).

[n^12: n in [0..30]]; // Vincenzo Librandi, May 06 2011

Range[0,30]^12 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

A008456(n)=n^12 \\ M. F. Hasler, Jul 03 2025

A008456 = lambda n: n**12 # M. F. Hasler, Jul 03 2025

Multiplicative with a(p^e) = p^(12*e). - David W. Wilson, Aug 01 2001
a(n) = A000290(n)^6 = A000578(n)^4 = A000583(n)^3 = A001014(n)^2. - Doug Bell, Jun 03 2017
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(12) = 691*Pi^12/638512875 (A013670).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2047*zeta(12)/2048 = 1414477*Pi^12/1307674368000. (End)
a(n) = 13*a(n-1)-78*a(n-2)+286*a(n-3)-715*a(n-4)+1287*a(n-5)-1716*a(n-6)+1716*a(n-7)-1287*a(n-8)+715*a(n-9)-286*a(n-10)+78*a(n-11)-13*a(n-12)+a(n-13). - Wesley Ivan Hurt, Dec 02 2021
Intersection of A000578 and A000583; i.e., cubes and 4th powers. - M. F. Hasler, Jul 03 2025

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

A244003 A(n,k) = k^Fibonacci(n); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 3, 4, 1, 0, 1, 5, 4, 9, 8, 1, 0, 1, 6, 5, 16, 27, 32, 1, 0, 1, 7, 6, 25, 64, 243, 256, 1, 0, 1, 8, 7, 36, 125, 1024, 6561, 8192, 1, 0, 1, 9, 8, 49, 216, 3125, 65536, 1594323, 2097152, 1, 0

Offset: 0

Alois P. Heinz, Jun 17 2014

			Square array A(n,k) begins:
  1, 1,   1,    1,     1,      1,       1, ...
  0, 1,   2,    3,     4,      5,       6, ...
  0, 1,   2,    3,     4,      5,       6, ...
  0, 1,   4,    9,    16,     25,      36, ...
  0, 1,   8,   27,    64,    125,     216, ...
  0, 1,  32,  243,  1024,   3125,    7776, ...
  0, 1, 256, 6561, 65536, 390625, 1679616, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0-10 give: A000007, A000012, A000301, A010098, A010099, A214706, A215270, A214887, A215271, A215272, A010100.
Rows n=0, 1+2, 3-8 give: A000012, A001477, A000290, A000578, A000584, A001016, A010801, A010809.
Main diagonal gives: A152915.
Cf. A000045, A103323.

A:= (n, k)-> k^(<<1|1>, <1|0>>^n)[1, 2]:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[0, 0] = 1; A[n_, k_] := k^Fibonacci[n]; Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 11 2015 *)

A(n,k) = k^A000045(n).

A(0,k) = 1, A(1,k) = k, A(n,k) = A(n-1,k) * A(n-2,k) for n>=2.

A138031 a(n) = prime(n)^13.

Original entry on oeis.org

8192, 1594323, 1220703125, 96889010407, 34522712143931, 302875106592253, 9904578032905937, 42052983462257059, 504036361936467383, 10260628712958602189, 24417546297445042591, 243569224216081305397, 925103102315013629321, 1718264124282290785243

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 01 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Subsequence of A010801 (13th powers).

Cf. A000040 (primes), A013671 (zeta(13)).

[p^13: p in PrimesUpTo(100)]; // Vincenzo Librandi, Mar 27 2014

Array[Prime[ # ]^13 &, 10]
Prime[Range[30]]^13 (* Vincenzo Librandi, Mar 27 2014 *)

A138031(n)=prime(n)^13 \\ M. F. Hasler, Jul 03 2025

From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(13)/zeta(26).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(13) = 1/A013671. (End)

More terms from Vincenzo Librandi, Mar 27 2014

A089081 26th powers: a(n) = n^26.

Original entry on oeis.org

0, 1, 67108864, 2541865828329, 4503599627370496, 1490116119384765625, 170581728179578208256, 9387480337647754305649, 302231454903657293676544, 6461081889226673298932241, 100000000000000000000000000

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Dec 04 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to divisibility sequences

Cf. A008454-A008456, A010801-A010813.

[n^26: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

Range[0, 9]^26 (* Alonso del Arte, Apr 26 2015 *)

a(n)=n^26 \\ Charles R Greathouse IV, Jun 28 2015

a(n) = n^26.
Completely multiplicative sequence with a(p) = p^26 for prime p. Multiplicative sequence with a(p^e) = p^(26e). - Jaroslav Krizek, Nov 01 2009
From Amiram Eldar, Oct 09 2020: (Start)
Dirichlet g.f.: zeta(s-26).
Sum_{n>=1} 1/a(n) = zeta(26) = 1315862*Pi^26/11094481976030578125.
Sum_{n>=1} (-1)^(n+1)/a(n) = 33554431*zeta(26)/33554432 = 22076500342261*Pi^26/186134520519971831808000000. (End)

A181134 Sum of 13th powers: a(n) = Sum_{j=0..n} j^13.

Original entry on oeis.org

0, 1, 8193, 1602516, 68711380, 1289414505, 14350108521, 111239118928, 660994932816, 3202860761145, 13202860761145, 47725572905076, 154718778284148, 457593884876401, 1251308658130545, 3197503726489920

Offset: 0

Bruno Berselli, Oct 05 2010 - Oct 18 2010

This form of recurrence is a general property of the array in A103438 (sums of the first n-th powers).

Bruno Berselli, Table of n, a(n) for n = 0..10000.
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
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. A010801.
Sequences of the form Sum_{j=0..n} j^m : A000217 (m=1), A000330 (m=2), A000537 (m=3), A000538 (m=4), A000539 (m=5), A000540 (m=6), A000541 (m=7), A000542 (m=8), A007487 (m=9), A023002 (m=10), A123095 (m=11), A123094 (m=12), A181134 (m=13).
Cf. A215083, A253712.

[(&+[j^13: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jul 21 2021

A181134 := proc(n) (bernoulli(14,n+1) - bernoulli(14))/14 ; end proc: seq(A181134(n), n=0..10); # R. J. Mathar, Oct 14 2010

Accumulate[Range[0,20]^13] (* Harvey P. Dale, Oct 30 2017 *)

A181134_list, m = [0], [6227020800, -37362124800, 97037740800, -142702560000, 130456085760, -76592355840, 28805736960, -6711344640, 901020120, -60780720, 1569750, -8190, 1, 0 , 0]
for _ in range(10**2):
    for i in range(14):
        m[i+1]+= m[i]
    A181134_list.append(m[-1]) # Chai Wah Wu, Nov 06 2014

[(bernoulli_polynomial(n+1, 14) - bernoulli(14))/14  for n in (0..30)] # G. C. Greubel, Jul 21 2021

For n>0, a(n) = n*A123094(n) - Sum_{i=0..n-1} A123094(i), where Sum_{i=0..n-1} A123094(i) = A253712(n-1) = (n-1)*n^2*(n+1)*(30*n^10 - 425*n^8 + 2578*n^6 - 8147*n^4 + 12874*n^2 - 7601)/5460.
a(n) = a(-n-1) = (n*(n + 1))^2*(30*n^10 + 150*n^9 + 125*n^8 - 400*n^7 - 326*n^6 + 1052*n^5 + 367*n^4 - 1786*n^3 + 202*n^2 + 1382*n - 691)/420.
G.f.: see comment of Vladeta Jovovic in A000538.
a(n) = -Sum_{j=1..13} j*Stirling1(n+1,n+1-j)*Stirling2(n+13-j,n). - Mircea Merca, Jan 25 2014

A022529 Nexus numbers (n+1)^13-n^13.

Original entry on oeis.org

1, 8191, 1586131, 65514541, 1153594261, 11839990891, 83828316391, 452866803481, 1992110014441, 7458134171671, 24522712143931, 72470493235141, 195881901213181, 490839666661891, 1152480295105231, 2557404559011121, 5400978405535441, 10918386832765231

Offset: 0

N. J. A. Sloane

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Column k=12 of array A047969.

Cf. A008292, A022528.

[(n+1)^13-n^13: n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

q=13;lst={};Do[AppendTo[lst,(n+1)^q-n^q],{n,0,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 23 2009 *)
Table[(n+1)^13-n^13,{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)

a(n) = (n+1)^13 - n^13; \\ Michel Marcus, Sep 25 2014

a(n) = A010801(n+1) - A010801(n). - Michel Marcus, Sep 25 2014
G.f.: -(x^12 +8178*x^11 +1479726*x^10 +45533450*x^9 +423281535*x^8 +1505621508*x^7 +2275172004*x^6 +1505621508*x^5 +423281535*x^4 +45533450*x^3 +1479726*x^2 +8178*x +1) / (x-1)^13. - Colin Barker, Sep 25 2014
G.f.: polylog(-13, x)*(1-x)/x. See the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021

cp's OEIS Frontend

A155016 Integer part of square root of A010801.

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A253713 Second partial sums of 13th powers (A010801).

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

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A008456 12th powers: a(n) = n^12.

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A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

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A244003 A(n,k) = k^Fibonacci(n); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A138031 a(n) = prime(n)^13.

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