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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

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).

Crossrefs

Cf. A000584 (5th powers), A013665 (zeta(7)), A275710 (eta(7)), A300785.

Cf. A003369 - A003379 (sums of 2, ..., 12 positive seventh powers).

Programs

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 *)

Maxima

makelist(n^7,n,0,20); /* Martin Ettl, Jan 15 2013 */

PARI

A001015(n)=n^7 \\ Charles R Greathouse IV, Sep 24 2015

Python

A001015 = lambda n: n**7 # M. F. Hasler, Jul 03 2025

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)

Extensions

More terms from James Sellers, Sep 19 2000