A001015 Seventh powers: a(n) = n^7. (Formerly M5392 N2341)

0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, 410338673, 612220032, 893871739, 1280000000, 1801088541, 2494357888, 3404825447, 4586471424, 6103515625, 8031810176

Offset

0,3

Comments

For n>0, (a(3*n-1)^7-a(2*n-1)^7-a(n)^7)/(7*(3*n-1)*(2*n-1)*n) = (2*A001106(n)+1)^2 (see Barisien reference, problem 173). - Bruno Berselli, Feb 01 2011

Number of the form a(n) + a(n+1) + ... + a(n+k) is never prime for all n, k>=0. This could be proved by the method indicated in the comment in A256581. - Vladimir Shevelev and Peter J. C. Moses, Apr 04 2015

References

E.-N. Barisien, Supplemento al Periodico di Matematica, Raffaello Giusti Editore (Livorno), July 1913, p. 135 (Problem 173).

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).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

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

Formula

Multiplicative with a(p^e) = p^(7e). - David W. Wilson, Aug 01 2001

Totally multiplicative sequence with a(p) = p^7 for primes p. - Jaroslav Krizek, Nov 01 2009

a(n) = 7*a(n-1) - 21* a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + 5040. - Ant King, Sep 24 2013

a(n) = n + Sum_{j=0..n-1}{k=1..6}binomial(7,k)*j^(7-k). - Patrick J. McNab, Mar 28 2016

G.f.: x*(1+120*x+1191*x^2+2416*x^3+1191*x^4+120*x^5+x^6)/(1-x)^8. See the Maple program. - Wolfdieter Lang, Oct 14 2016

From Kolosov Petro, Oct 22 2018: (Start)

a(n) = Sum_{k=1..n} A300785(n,k).

a(n) = Sum_{k=0..n-1} A300785(n,k). (End)

From Amiram Eldar, Oct 08 2020: (Start)

Sum_{n>=1} 1/a(n) = zeta(7) (A013665).

Sum_{n>=1} (-1)^(n+1)/a(n) = 63*zeta(7)/64 (A275710). (End)

Maple

A001015:=z*(1191*z^4+120*z^5+1191*z^2+2416*z^3+120*z+z^6+1)/(z-1)^8; # Simon Plouffe in his 1992 dissertation; offset corrected by M. F. Hasler, Feb 01 2011

Mathematica

Table[n^7, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

Prog

(Maxima) makelist(n^7, n, 0, 20); /* Martin Ettl, Jan 15 2013 */
(PARI) a(n)=n^7 \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. A000584, A013665, A275710, A300785.

Sequence in context: A250365 A017678 A123253 * A352053 A050754 A351605

Adjacent sequences: A001012 A001013 A001014 * A001016 A001017 A001018

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Sep 19 2000

Status

approved