A002604 -id:A002604 - cp's OEIS frontend

A001014 Sixth powers: a(n) = n^6.

0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 244140625, 308915776, 387420489, 481890304

Offset: 0

N. J. A. Sloane

Numbers both square and cubic. - Patrick De Geest
Totally multiplicative sequence with a(p) = p^6 for prime p. - Jaroslav Krizek, Nov 01 2009
Numbers n for which the order of the torsion subgroup of the elliptic curve y^2 = x^3 + n is t = 6, cf. Gebel link. - Artur Jasinski, Jun 30 2010
Note that Sum_{n>=1} 1/a(n) = Pi^6 / 945. - Mohammad K. Azarian, Nov 01 2011
The binomial transform yields A056468. The inverse binomial transform yields the (finite) 0, 1, 62, 540, ..., 720, the 6th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
For n > 0, a(n) is the largest number k such that k + n^3 divides k^2 + n^3. - Derek Orr, Oct 01 2014

			The 6th powers of the first few integers are: 0^6 = 0 = a(0), 1^6 = 1 = a(1), 2^6 = 64 = a(2), 3^6 = 9^3 = 729 = a(3), 4^6 = 2^12 = 4096 = a(4), 5^6 = 25^3 = 15625 = a(5), etc.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity, eq. (6.37).
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).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..500
Henry Bottomley, Illustration of initial terms
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]
Richard J. Mathar, Construction of Bhaskara pairs, arXiv:1703.01677 [math.NT], 2017.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Subsequence of A201217.
Cf. A000540 (partial sums), A022522 (first differences), A008292.
Intersection of A000290 (squares) and A000578 (cubes).
Cf. A002604 (n^6+1), A123866 (n^6-1), A013664 (zeta(6)), A275703 (eta(6)).
Cf. A003358 - A003368 (sums of 2, ..., 12 positive sixth powers).

a001014 n = a001014_list !! n
a001014_list = map (^ 6) [0..] -- Reinhard Zumkeller, Dec 04 2011

Table[n^6, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

A001014(n):=n^6$
makelist(A001014(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

A001014(n)=n^6 \\ Charles R Greathouse IV, Sep 24 2015

A001014 = lambda n: n**6 # M. F. Hasler, Jul 03 2025

a(n) = A123866(n) + 1 = A002604(n) - 1.
G.f.: -x*(1+x)*(x^4+56*x^3+246*x^2+56*x+1) / (x-1)^7. - Simon Plouffe in his 1992 dissertation
Multiplicative with a(p^e) = p^(6e). - David W. Wilson, Aug 01 2001
E.g.f.: (x + 31x^2 + 90x^3 + 65x^4 + 15x^5 + x^6)*exp(x).  Generally, the e.g.f. for n^m is Sum_{k=1..m} A008277(m,k)*x^k*exp(x). - Geoffrey Critzer, Aug 25 2013
From Ant King, Sep 23 2013: (Start)
Signature {7, -21, 35, -35, 21, -7, 1}.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + 720. (End)
a(n) == 1 (mod 7) if gcd(n, 7) = 1, otherwise a(n) == 0 (mod 7). See A109720. - Jake Lawrence, May 28 2016
From Ilya Gutkovskiy, Jul 06 2016: (Start)
Dirichlet g.f.: zeta(s-6).
Sum_{n>=1} 1/a(n) = Pi^6/945 = A013664. (End)
a(n) = Sum_{k=1..6} Eulerian(6, k)*binomial(n+6-k, 6), with Eulerian(6, k) = A008292(6, k) (the numbers are 1, 57, 302, 302, 57, 1) for n >= 0. Worpitzki's identity for powers of 6. See. e.g., Graham et al., eq. (6, 37) (using A173018, the row reversed version of A123125). - Wolfdieter Lang, Jul 17 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 31*zeta(6)/32 = 31*Pi^6/30240 (A275703). - Amiram Eldar, Oct 08 2020
From Amiram Eldar, Jan 20 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = (cosh(Pi)-cos(sqrt(3)*Pi))*sinh(Pi)/(2*Pi^3).
Product_{n>=2} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)^2/(6*Pi^2). (End)

Comments from 2010 - 2011 edited by M. F. Hasler, Jul 05 2024

A060890 a(n) = n^8 + 1.

Original entry on oeis.org

1, 2, 257, 6562, 65537, 390626, 1679617, 5764802, 16777217, 43046722, 100000001, 214358882, 429981697, 815730722, 1475789057, 2562890626, 4294967297, 6975757442, 11019960577, 16983563042, 25600000001, 37822859362, 54875873537, 78310985282, 110075314177, 152587890626

Offset: 0

N. J. A. Sloane, May 05 2001

a(n) = Phi_16(n) where Phi_k(x) is the k-th cyclotomic polynomial.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A002522, A001093, A002523, A002561, A002604.

A060890 := proc(n)
        numtheory[cyclotomic](16,n) ;
end proc:
seq(A060890(n),n=0..20) ; # R. J. Mathar, Feb 11 2014

Table[n^8+1,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,2,257,6562,65537,390626,1679617,5764802,16777217},40] (* Harvey P. Dale, Mar 12 2013 *)

a(n) = n^8 + 1 \\ Harry J. Smith, Jul 14 2009

a(0)=1, a(1)=2, a(2)=257, a(3)=6562, a(4)=65537, a(5)=390626, a(6)=1679617, a(7)=5764802, a(8)=16777217, 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). - Harvey P. Dale, Mar 12 2013
Sum_{n>=0} 1/a(n) = 1/2 + Pi*((sqrt(2 + sqrt(2)) * sin(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2)) * sinh(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 - sqrt(2))*Pi) - cos(sqrt(2 + sqrt(2))*Pi)) + (sqrt(2 - sqrt(2)) * sin(sqrt(2 - sqrt(2))*Pi) + sqrt(2 + sqrt(2)) * sinh(sqrt(2 + sqrt(2))*Pi)) / (cosh(sqrt(2 + sqrt(2))*Pi) - cos(sqrt(2 - sqrt(2))*Pi))) / 8 = 1.5040621333147995112929... . - Vaclav Kotesovec, Feb 14 2015
Sum_{n>=0} (-1)^n/a(n) = 1/2 + Pi*((sqrt(2 - sqrt(2)) * sin(sqrt(2 - sqrt(2))*Pi/2) - sqrt(2 + sqrt(2)) * sinh(sqrt(2 + sqrt(2))*Pi/2)) / (cos(sqrt(2 - sqrt(2))*Pi/2) + cosh(sqrt(2 + sqrt(2))*Pi/2)) - (sqrt(2 - sqrt(2)) * sin(sqrt(2 - sqrt(2))*Pi/2) + sqrt(2 + sqrt(2)) * sinh(sqrt(2 + sqrt(2))*Pi/2)) / (cos(sqrt(2 - sqrt(2))*Pi/2) - cosh(sqrt(2 + sqrt(2))*Pi/2)) + (sqrt(2 + sqrt(2)) * sin(sqrt(2 + sqrt(2))*Pi/2) - sqrt(2 - sqrt(2)) * sinh(sqrt(2 - sqrt(2))*Pi/2)) / (cos(sqrt(2 + sqrt(2))*Pi/2) + cosh(sqrt(2 - sqrt(2))*Pi/2)) - (sqrt(2 + sqrt(2)) * sin(sqrt(2 + sqrt(2))*Pi/2) + sqrt(2 - sqrt(2)) * sinh(sqrt(2 - sqrt(2))*Pi/2)) / (cos(sqrt(2 + sqrt(2))*Pi/2) - cosh(sqrt(2 - sqrt(2))*Pi/2)))/16 = 0.5037518217314416642671664241... . - Vaclav Kotesovec, Feb 14 2015
G.f.: (1-7*x+275*x^2+4237*x^3+15689*x^4+15563*x^5+4321*x^6+239*x^7+2*x^8)/ (1-x)^9. - Colin Barker, Apr 21 2012

A258806 a(n) = n^7 + 1.

Original entry on oeis.org

1, 2, 129, 2188, 16385, 78126, 279937, 823544, 2097153, 4782970, 10000001, 19487172, 35831809, 62748518, 105413505, 170859376, 268435457, 410338674, 612220033, 893871740, 1280000001, 1801088542, 2494357889, 3404825448, 4586471425, 6103515626, 8031810177

Offset: 0

Vincenzo Librandi, Jun 11 2015

Muniru A Asiru, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Subsequence of A004864.
Sequences of the type n^k+1: A002522 (k=2), A001093 (k=3), A002523 (k=4), A002561 (k=5), A002604 (k=6), this sequence (k=7), A060890 (k=8).
Cf. A300785.

List([0..30],n->n^7+1); # Muniru A Asiru, Oct 24 2018

Magma
```
[n^7+1: n in [0..40]];
```

I:=[1,2,129,2188, 16385,78126,279937,823544]; [n le 8 select I[n] else 8*Self(n-1) - 28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5) -28*Self(n-6) + 8*Self(n-7)-Self(n-8): n in [1..40]];

seq(n^7+1,n=0..30); # Muniru A Asiru, Oct 24 2018

Table[n^7 + 1, {n, 0, 40}] (* or *) LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {1, 2, 129, 2188, 16385, 78126, 279937, 823544}, 40]

a(n)=n^7+1 \\ Charles R Greathouse IV, Jun 11 2015

[n^7+1 for n in (1..40)] # Bruno Berselli, Jun 11 2015

G.f.: (1 - 6*x + 141*x^2 + 1156*x^3 + 2451*x^4 + 1170*x^5 + 127*x^6)/(1 - x)^8.
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).
a(n) = (n + 1)*(n^6 - n^5 + n^4 - n^3 + n^2 - n + 1).
a(n) = Sum_{k=0..n} A300785(n,k). - Kolosov Petro, Oct 23 2018
E.g.f.: (1 +x +63*x^2 +301*x^3 +350*x^4 +140*x^5 +*21*x^6 +x^7)*exp(x). - G. C. Greubel, Oct 24 2018

A354052 Decimal expansion of Sum_{k>=0} 1 / (k^6 + 1).

Original entry on oeis.org

1, 5, 1, 7, 1, 0, 0, 7, 3, 4, 0, 3, 3, 2, 1, 6, 4, 2, 6, 1, 5, 2, 9, 0, 7, 6, 4, 4, 9, 0, 2, 4, 1, 3, 8, 5, 8, 0, 6, 2, 2, 1, 1, 3, 2, 2, 5, 2, 9, 8, 4, 4, 6, 7, 2, 8, 4, 7, 6, 3, 4, 8, 9, 9, 0, 3, 7, 9, 0, 1, 3, 5, 0, 5, 3, 5, 7, 9, 8, 7, 2, 0, 0, 7, 8, 4, 3, 6, 9, 3, 6, 9, 3, 3, 0, 0, 6, 4, 3, 7, 0, 6, 6, 6, 4

Offset: 1

Vaclav Kotesovec, May 16 2022

			1.517100734033216426152907644902413858062211322529844672847634899037901...

Cf. A002604, A113319, A354051, A354053.

evalf(1/2 + (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi)))*Pi/6, 100);

RealDigits[Chop[N[Sum[1/(k^6 + 1), {k, 0, Infinity}], 105]]][[1]]

PARI
```
sumpos(k=0, 1/(k^6 + 1))
```

Equals 1/2 + (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi)))*Pi/6.

Equal 3/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(6*k)-1). - Amiram Eldar, May 20 2022

A354054 a(n) = Product_{k=0..n} (k^6 + 1).

Original entry on oeis.org

1, 2, 130, 94900, 388805300, 6075471617800, 283463279271694600, 33349454806314869690000, 8742392830201411514885050000, 4646074730467898538293540742100000, 4646079376542629006192079035640742100000, 8230817672466612927467651920537784356160200000

Offset: 0

Vaclav Kotesovec, May 16 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..103

Cf. A002604, A101686, A255433, A255434, A258871, A354052.

A354054 := proc(n)
    mul( k^6+1,k=0..n) ;
end proc:
seq(A354054(n),n=0..40) ; # R. J. Mathar, Jul 17 2023

Table[Product[k^6 + 1, {k, 0, n}], {n, 0, 15}]

PARI
```
a(n) = prod(k=1, n, k^6+1);
```

a(n) ~ (2*sinh(2*Pi) - 4*sinh(Pi)*cos(sqrt(3)*Pi)) * n^(6*n + 3) / exp(6*n).

a(n) ~ A258871 * (n!)^6.

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A001014 Sixth powers: a(n) = n^6.

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A060890 a(n) = n^8 + 1.

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A354052 Decimal expansion of Sum_{k>=0} 1 / (k^6 + 1).

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A354054 a(n) = Product_{k=0..n} (k^6 + 1).

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