A008456 -id:A008456 - cp's OEIS frontend

A010804 16th powers: a(n) = n^16.

0, 1, 65536, 43046721, 4294967296, 152587890625, 2821109907456, 33232930569601, 281474976710656, 1853020188851841, 10000000000000000, 45949729863572161, 184884258895036416, 665416609183179841, 2177953337809371136, 6568408355712890625, 18446744073709551616, 48661191875666868481

Offset: 0

N. J. A. Sloane

Exponent towers of the form n^(2^(2^2)). - Paul Duckett, Aug 30 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. A013674 (zeta(16)).

Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A001016 (8th powers), A008456 (12th powers).

[n^16: n in [0..15]]; // Vincenzo Librandi, Jun 19 2011

A010804 := n -> n^16; # M. F. Hasler, Jul 03 2025

Range[0, 15]^16 (* Alonso del Arte, Feb 16 2015 *)

A010804(n):=n^16$
makelist(A010804(n),n,0,10); /* Martin Ettl, Nov 12 2012 */

A010804(n)=n^16 \\ Charles R Greathouse IV, Jun 28 2015

A010804 = lambda n: n**16 # M. F. Hasler, Jul 03 2025

Completely multiplicative with a(p) = p^16 for prime p. Multiplicative with a(p^e) = p^(16e). - Jaroslav Krizek, Nov 01 2009
From Ilya Gutkovskiy, Feb 27 2017: (Start)
Dirichlet g.f.: zeta(s-16).
Sum_{n>=1} 1/a(n) =  3617*Pi^16/325641566250 = A013674. (End)
a(n) = A001016(n)^2. - Michel Marcus, Feb 28 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = 32767*zeta(16)/32768 = 16931177*Pi^16/1524374691840000. - Amiram Eldar, Oct 08 2020

A036090 Centered cube numbers: (n+1)^12 + n^12.

Original entry on oeis.org

1, 4097, 535537, 17308657, 260917841, 2420922961, 16018069537, 82560763937, 351149013217, 1282429536481, 4138428376721, 12054528824977, 32214185570737, 79991997497777, 186440250265921, 411221314601281

Offset: 0

N. J. A. Sloane

nonn

Never prime, as a(n) = (2n^4 + 4n^3 + 6n^2 + 4n + 1) * (n^8 + 4n^7 + 22n^6 + 52n^5 + 69n^4 + 56n^3 + 28n^2 + 8n + 1) Semiprime for n in {1, 2, 3, 6, 14, 16, 36, 87, 97, 109, 110, 119, 121, 163, 195, ...}. - Jonathan Vos Post, Aug 26 2011

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A008456, A036088, A036089, A100267, A154535, A100266, A152913, A194155, A194185, A194216.

[(n+1)^12+n^12: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

Total/@Partition[Range[0,20]^12,2,1] (* Harvey P. Dale, May 09 2018 *)

G.f.: -(x^10 + 4082*x^9 + 474189*x^8 + 9713496*x^7 + 56604978*x^6 + 105907308*x^5 + 56604978*x^4 + 9713496*x^3 + 474189*x^2 + 4082*x + 1)*(1+x)^2 / (x-1)^13. - R. J. Mathar, Aug 27 2011

A123094 Sum of first n 12th powers.

Original entry on oeis.org

0, 1, 4097, 535538, 17312754, 261453379, 2438235715, 16279522916, 84998999652, 367428536133, 1367428536133, 4505856912854, 13421957361110, 36720042483591, 93413954858887, 223160292749512, 504635269460168, 1087257506689929, 2244088888116105, 4457403807182266

Offset: 0

Zerinvary Lajos, Sep 27 2006

T. D. Noe, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Formula lines (first formula): website Matem@ticamente (in Italian).

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), this sequence (m=12), A181134 (m=13).
Cf. A008456, A215083.
Cf. A008275, A008277.

[(&+[j^12: j in [0..n]]): j in [0..30]]; // G. C. Greubel, Jul 21 2021

Maple
```
[seq(add(i^12, i=1..n), n=0..18)];
```

Table[Sum[k^12, {k, n}], {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0,30]^12]  (* Harvey P. Dale, Apr 26 2011 *)

A123094_list, m = [0], [479001600, -2634508800, 6187104000, -8083152000, 6411968640, -3162075840, 953029440, -165528000, 14676024, -519156, 4094, -1, 0 , 0]
for _ in range(10**2):
    for i in range(13):
        m[i+1]+= m[i]
    A123094_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

[bernoulli_polynomial(n,13)/13 for n in range(1, 30)] # Zerinvary Lajos, May 17 2009

a(n) = n*A123095(n) - Sum_{i=0..n-1} A123095(i). - Bruno Berselli, Apr 27 2010
a(n) = n * (n+1) * (2*n+1) * (105*n^10 +525*n^9 +525*n^8 -1050*n^7 -1190*n^6 +2310*n^5 +1420*n^4 -3285*n^3 -287*n^2 +2073*n -691)/2730. - Bruno Berselli, Oct 03 2010
a(n) = (-1)*Sum_{j=1..12} j*Stirling1(n+1,n+1-j)*Stirling2(n+12-j,n). - Mircea Merca, Jan 25 2014

A016788 a(n) = (3*n+1)^12.

Original entry on oeis.org

1, 16777216, 13841287201, 1000000000000, 23298085122481, 281474976710656, 2213314919066161, 12855002631049216, 59604644775390625, 232218265089212416, 787662783788549761, 2386420683693101056, 6582952005840035281, 16777216000000000000, 39959630797262576401

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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A008456, A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784, A016785, A016786, A016787.

Subsequence of A008456.

[(3*n+1)^12: n in [0..20]]; // Vincenzo Librandi, Sep 29 2011

Table[(3n+1)^12,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)

a(n) = A008456(A016777(n)). - Michel Marcus, Jun 16 2016

Sum_{n>=0} 1/a(n) = PolyGamma(11, 1/3)/21213424108800. - Amiram Eldar, Mar 30 2022

A071235 a(n) = (n^12 + n^6)/2.

Original entry on oeis.org

0, 1, 2080, 266085, 8390656, 122078125, 1088414496, 6920702425, 34359869440, 141215033961, 500000500000, 1569215074141, 4458051717120, 11649044974645, 28346959952416, 64873174640625, 140737496743936, 291311130683665, 578415707719200, 1106657483056021

Offset: 0

N. J. A. Sloane, Jun 12 2002

Number of unoriented rows of length 12 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=2080, there are 2^12=4096 oriented arrangements of two colors. Of these, 2^6=64 are achiral. That leaves (4096-64)/2=2016 chiral pairs. Adding achiral and chiral, we get 2080. - Robert A. Russell, Nov 13 2018

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Row 12 of A277504.

Cf. A008456 (oriented), A001014 (achiral).

List([0..40], n -> (n^12 + n^6)/2); # G. C. Greubel, Nov 15 2018

[n^6*(n^2+1)*(n^4-n^2+1)/2: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

Table[(n^12 + n^6)/2, {n,0,30}] (* Robert A. Russell, Nov 13 2018 *)

vector(40, n, n--; ) \\ G. C. Greubel, Nov 15 2018

for n in range(0,20): print(int((n**12 + n**6)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^6*(1 + n^6)/2 for n in range(40)] # G. C. Greubel, Nov 15 2018

a(n) = n^6*(n^2 + 1)*(n^4 - n^2 + 1)/2.
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A008456(n) + A001014(n)) / 2 = (n^12 + n^6) / 2.
G.f.: (Sum_{j=1..12} S2(12,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..6} S2(6,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..11} A145882(12,k) * x^k / (1-x)^13.
E.g.f.: (Sum_{k=1..12} S2(12,k)*x^k + Sum_{k=1..6} S2(6,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>12, a(n) = Sum_{j=1..13} -binomial(j-14,j) * a(n-j). (End)
From G. C. Greubel, Nov 15 2018: (Start)
G.f.: x*(1 +2067*x +239123*x^2 +5093505*x^3 +33160062*x^4 + 81255642*x^5 +81255642*x^6 +33160062*x^7 +5093505*x^8 +239123*x^9 +2067*x^10 +x^11)/( 1-x)^13.
E.g.f.: x*(2 +2078*x +86616*x^2 +611566*x^3 +1379415*x^4 +*1323653*x^5 + 627396*x^6 +159027*x^7 +22275*x^8 +1705*x^9 +66*x^10 +x^11)*exp(x)/2. (End)

New name from G. C. Greubel, Nov 15 2018

A016776 a(n) = (3*n)^12.

Original entry on oeis.org

0, 531441, 2176782336, 282429536481, 8916100448256, 129746337890625, 1156831381426176, 7355827511386641, 36520347436056576, 150094635296999121, 531441000000000000, 1667889514952984961, 4738381338321616896

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 (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A008456 (n^12).

[(3*n)^12: n in [0..20]]; // Vincenzo Librandi, May 09 2011

(3Range[0,20])^12 (* Harvey P. Dale, Apr 25 2011 *)

A016800 a(n) = (3*n + 2)^12.

Original entry on oeis.org

4096, 244140625, 68719476736, 3138428376721, 56693912375296, 582622237229761, 4096000000000000, 21914624432020321, 95428956661682176, 353814783205469041, 1152921504606846976, 3379220508056640625, 9065737908494995456, 22563490300366186081, 52654090776777588736

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016789, A016790, A016791, A016792, A016793, A016794, A016795, A016796, A016797, A016798, A016799.

Subsequence of A008456.

Table[(3n+2)^12,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)

From Amiram Eldar, Apr 01 2022: (Start)
a(n) = A016789(n)^12 = A016790(n)^6 = A016791(n)^4 = A016792(n)^3 = A016794(n)62.
Sum_{n>=0} 1/a(n) = PolyGamma(11, 2/3)/21213424108800. (End)

A016848 a(n) = (4*n+3)^12.

Original entry on oeis.org

531441, 13841287201, 3138428376721, 129746337890625, 2213314919066161, 21914624432020321, 150094635296999121, 787662783788549761, 3379220508056640625, 12381557655576425121, 39959630797262576401

Offset: 0

N. J. A. Sloane

David A. Corneth, Table of n, a(n) for n = 0..9999
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A008456.

Cf. A004767, A016838, A016839, A016840, A016841, A016842, A016843, A016844, A016845, A016846, A016847.

List([0..12], n-> (4*n+3)^12); # G. C. Greubel, Aug 29 2019

[(4*n+3)^12: n in [0..12]]; // G. C. Greubel, Aug 29 2019

seq((4*n+3)^12, n=0..12); # G. C. Greubel, Aug 29 2019

(4*Range[13]-1)^12 (* G. C. Greubel, Aug 29 2019 *)

a(n) = (4*n+3)^12 \\ David A. Corneth, Aug 28 2019

[(4*n+3)^12 for n in (0..12)] # G. C. Greubel, Aug 29 2019

A016920 a(n) = (6*n)^12.

Original entry on oeis.org

0, 2176782336, 8916100448256, 1156831381426176, 36520347436056576, 531441000000000000, 4738381338321616896, 30129469486639681536, 149587343098087735296, 614787626176508399616, 2176782336000000000000

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 (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A008456, A008588.

[(6*n)^12: n in [0..25]]; // Vincenzo Librandi, May 03 2011

(6*Range[0,20])^12 (* or *) LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{0,2176782336,8916100448256,1156831381426176,36520347436056576,531441000000000000,4738381338321616896,30129469486639681536,149587343098087735296,614787626176508399616,2176782336000000000000,6831675453247426400256,19408409961765342806016},20] (* Harvey P. Dale, Mar 17 2019 *)

A016944 a(n) = (6*n + 2)^12.

Original entry on oeis.org

4096, 68719476736, 56693912375296, 4096000000000000, 95428956661682176, 1152921504606846976, 9065737908494995456, 52654090776777588736, 244140625000000000000, 951166013805414055936, 3226266762397899821056, 9774779120406941925376, 26963771415920784510976

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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016788, A016933, A016934, A016935, A016936, A016937, A016938, A016939, A016940, A016941, A016942, A016943.

Subsequence of A008456.

[(6*n+2)^12: n in [0..20]]; // Vincenzo Librandi, May 05 2011

(6*Range[0,20]+2)^12 (* or *) LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{4096,68719476736,56693912375296,4096000000000000,95428956661682176,1152921504606846976,9065737908494995456,52654090776777588736,244140625000000000000,951166013805414055936,3226266762397899821056,9774779120406941925376,26963771415920784510976},20] (* Harvey P. Dale, Aug 03 2021 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016933(n)^12 = A016934(n)^6 = A016935(n)^4 = A016936(n)^3 = A016938(n)^2.
a(n) = 2^12*A016788(n).
Sum_{n>=0} 1/a(n) = PolyGamma(11, 1/3)/86890185149644800. (End)

cp's OEIS Frontend

A010804 16th powers: a(n) = n^16.

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A036090 Centered cube numbers: (n+1)^12 + n^12.

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A123094 Sum of first n 12th powers.

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A016788 a(n) = (3*n+1)^12.

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A071235 a(n) = (n^12 + n^6)/2.

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A016776 a(n) = (3*n)^12.

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A016800 a(n) = (3*n + 2)^12.

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