A008455 -id:A008455 - cp's OEIS frontend

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

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

A010801 13th powers: a(n) = n^13.

Original entry on oeis.org

0, 1, 8192, 1594323, 67108864, 1220703125, 13060694016, 96889010407, 549755813888, 2541865828329, 10000000000000, 34522712143931, 106993205379072, 302875106592253, 793714773254144, 1946195068359375, 4503599627370496, 9904578032905937, 20822964865671168

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 (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A000584 (5th powers), A008455 (11th powers), A013671 (zeta(11)).

[n^13: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

A010801 := n -> n^13; \\ M. F. Hasler, Jul 03 2025

Range[0,30]^13 (* Vladimir Joseph Stephan Orlovsky, Mar 14 2011 *)

A010801(n)=n^13 \\ Charles R Greathouse IV, Oct 07 2015

A010801 = lambda n: n**13 # M. F. Hasler, Jul 03 2025

a(n) mod 10 = n mod 10. - Reinhard Zumkeller, Dec 06 2004
Totally multiplicative with a(p) = p^13 for primes p. Multiplicative with a(p^e) = p^(13*e). - Jaroslav Krizek, Nov 01 2009
G.f.: x*(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)^14. - Colin Barker, Sep 25 2014
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(13) (A013671).
Sum_{n>=1} (-1)^(n+1)/a(n) = 4095*zeta(13)/4096. (End)

A079395 a(n) = prime(n)^11.

Original entry on oeis.org

2048, 177147, 48828125, 1977326743, 285311670611, 1792160394037, 34271896307633, 116490258898219, 952809757913927, 12200509765705829, 25408476896404831, 177917621779460413, 550329031716248441

Offset: 1

Jon Perry, Jan 06 2003

			2^11 = 2048.

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

Subsequence of A008455.

Cf. A013669.

[p^11: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A079395:=n->ithprime(n)^11: seq(A079395(n), n=1..30); # Wesley Ivan Hurt, Feb 07 2017

Array[Prime[ # ]^11 &, 30] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Prime[Range[20]]^11 (* Vincenzo Librandi, Mar 27 2014 *)

PARI
```
forprime (p=2,100,print1(p^11","))
```

From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(11)/zeta(22).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(11) = 1/A013669. (End)

A010803 15th powers: a(n) = n^15.

Original entry on oeis.org

0, 1, 32768, 14348907, 1073741824, 30517578125, 470184984576, 4747561509943, 35184372088832, 205891132094649, 1000000000000000, 4177248169415651, 15407021574586368, 51185893014090757, 155568095557812224

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 (16, -120, 560, -1820, 4368, -8008, 11440, -12870, 11440, -8008, 4368, -1820, 560, -120, 16, -1).

Cf. A013673 (zeta(15)).

Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A000584 (5th powers), A001015 (7th powers), A008455 (11th powers).

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

Table[n^15,{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2010 *)

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

A010803(n)=n^15 \\ M. F. Hasler, Jul 03 2025

A010803 = lambda n: n**15 # M. F. Hasler, Jul 03 2025

Totally multiplicative with a(p) = p^15 for prime p. Multiplicative with a(p^e) = p^(15e). - Jaroslav Krizek, Nov 01 2009
From Ilya Gutkovskiy, Feb 27 2017: (Start)
Dirichlet g.f.: zeta(s-15).
Sum_{n>=1} 1/a(n) = zeta(15) = A013673. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 16383*zeta(15)/16384. - Amiram Eldar, Oct 08 2020

A022527 Nexus numbers: a(n) = (n+1)^11 - n^11.

Original entry on oeis.org

1, 2047, 175099, 4017157, 44633821, 313968931, 1614529687, 6612607849, 22791125017, 68618940391, 185311670611, 457696700077, 1049152023349, 2257404775627, 4600190689711, 8942430185041, 16679710263217, 29996513771599, 52221848818987, 88309741101781

Offset: 0

N. J. A. Sloane

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

T. D. Noe, Table of n, a(n) for n = 0..1000
H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Column k=10 of A047969.
Cf. A008455 (n^11).
Cf. A008292, A022526, A022538.

[(n+1)^11-n^11: n in [0..40]]; // Vincenzo Librandi, Jan 26 2011

b:=11: a:=n->(n+1)^b-n^b: seq(a(n),n=0..18); # Muniru A Asiru, Feb 28 2018

q=11; Table[(n+1)^q-n^q, {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Jan 23 2009 *)
Differences[Range[0, 20]^11] (* Harvey P. Dale, Jan 25 2011 *)

for(n=0,20, print1((n+1)^11 - n^11, ", ")) \\ G. C. Greubel, Feb 27 2018

G.f.: -(x^10 + 2036*x^9 + 152637*x^8 + 2203488*x^7 + 9738114*x^6 + 15724248*x^5 + 9738114*x^4 + 2203488*x^3 + 152637*x^2 + 2036*x + 1) / (x - 1)^11. - Colin Barker, Dec 22 2012
a(n) = A008455(n+1) - A008455(n). - Michel Marcus, Feb 28 2018
G.f.: polylog(-11, x)*(1-x)/x. See the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021

A211184 Numbers k such that (k+1)^11 - k^11 is prime.

Original entry on oeis.org

5, 7, 9, 13, 34, 40, 63, 69, 85, 168, 170, 183, 207, 223, 247, 275, 291, 306, 322, 337, 344, 352, 381, 391, 397, 400, 404, 469, 473, 492, 570, 574, 579, 590, 597, 673, 680, 696, 736, 764, 786, 805, 827, 890, 915, 947, 1006, 1023, 1025, 1039

Offset: 1

Vladimir Pletser, Feb 02 2013

Vladimir Pletser, Table of n, a(n) for n = 1..1000

Cf. A008455 (11th powers), A189055 (resulting primes).

Select[Range[1000], PrimeQ[(# + 1)^11 - #^11] &] (* T. D. Noe, Feb 04 2013 *)

isok(k) = isprime((k+1)^11 - k^11); \\ Michel Marcus, Mar 12 2022

A016787 a(n) = (3*n + 1)^11.

Original entry on oeis.org

1, 4194304, 1977326743, 100000000000, 1792160394037, 17592186044416, 116490258898219, 584318301411328, 2384185791015625, 8293509467471872, 25408476896404831, 70188843638032384, 177917621779460413, 419430400000000000, 929293739471222707, 1951354384207722496

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784, A016785, A016786.

Subsequence of A008455.

[(3*n+1)^11: n in [0..20]]; // Vincenzo Librandi, Sep 29 2011

Table[(3*n + 1)^11, {n, 0, 30}] (* Amiram Eldar, Mar 30 2022 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016777(n)^11.
Sum_{n>=0} 1/a(n) = 7388*Pi^11/(2511058725*sqrt(3)) + 88573*zeta(11)/177147. (End)
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12). - Wesley Ivan Hurt, Apr 12 2023

A036089 Centered cube numbers: (n+1)^11 + n^11.

Original entry on oeis.org

1, 2049, 179195, 4371451, 53022429, 411625181, 2340123799, 10567261335, 39970994201, 131381059609, 385311670611, 1028320041299, 2535168764725, 5841725563701, 12699321029039, 26241941903791, 51864082352049

Offset: 0

N. J. A. Sloane

Never prime, as a(n) = (2*n+1) * (n^10 + 5*n^9 + 25*n^8 + 70*n^7 + 130*n^6 + 166*n^5 + 148*n^4 + 91*n^3 + 37*n^2 + 9*n + 1). - Jonathan Vos Post, Aug 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A008455, A036087, A036088, A100267, A154535, A100266, A152913, A194155, A194185, A194216.

[(n+1)^11+n^11: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

Vec((1 + x)*(1 + 2036*x + 152637*x^2 + 2203488*x^3 + 9738114*x^4 + 15724248*x^5 + 9738114*x^6 + 2203488*x^7 + 152637*x^8 + 2036*x^9 + x^10) / (1 - x)^12 + O(x^40)) \\ Colin Barker, Feb 06 2020

From Colin Barker, Feb 06 2020: (Start)
G.f.: (1 + x)*(1 + 2036*x + 152637*x^2 + 2203488*x^3 + 9738114*x^4 + 15724248*x^5 + 9738114*x^6 + 2203488*x^7 + 152637*x^8 + 2036*x^9 + x^10) / (1 - x)^12.
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12) for n>11.
(End)

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

cp's OEIS Frontend

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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A010801 13th powers: a(n) = n^13.

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A079395 a(n) = prime(n)^11.

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A010803 15th powers: a(n) = n^15.

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A022527 Nexus numbers: a(n) = (n+1)^11 - n^11.

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