A256581 -id:A256581 - cp's OEIS frontend

A000584 Fifth powers: a(n) = n^5.

0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 9765625, 11881376, 14348907, 17210368, 20511149

Offset: 0

N. J. A. Sloane

Totally multiplicative sequence with a(p) = p^5 for prime p. - Jaroslav Krizek, Nov 01 2009
The binomial transform yields A059338. The inverse binomial transform yields the (finite) 0, 1, 30, 150, 240, 120, the 5th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
Equals sum of odd numbers from n^2*(n-1)+1 (A100104) to n^2*(n+1)-1 (A003777). - Bruno Berselli, Mar 14 2014
a(n) mod 10 = n mod 10. - Reinhard Zumkeller, May 10 2014
Numbers of the form a(n) + a(n+1) + ... + a(n+k) are nonprime for all n, k>=0; this can be proved by the method indicated in the comment in A256581. - Vladimir Shevelev and Peter J. C. Moses, Apr 04 2015

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 (6.37).
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
Brady Haran and Simon Pampena, Fifth Root Trick, Numberphile video (2014)
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 (6,-15,20,-15,6,-1).

Partial sums give A000539.
Cf. A000012, A001477, A000290, A000578, A000583, A062392, A022521 (first differences), A008292, A173018, A123125.
Cf. A013663, A162624, A267316.

a000584 = (^ 5)  -- Reinhard Zumkeller, Nov 11 2012

Range[0,50]^5 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

makelist(n^5, n, 0, 30); /* Martin Ettl, Nov 12 2012 */

a(n)=n^5 \\ Charles R Greathouse IV, Jul 28 2015

[n**5 for n in range(30)]  # Zerinvary Lajos, Jun 03 2009

G.f.: x*(1+26*x+66*x^2+26*x^3+x^4) / (x-1)^6. [Simon Plouffe in his 1992 dissertation]
Multiplicative with a(p^e) = p^(5e). - David W. Wilson, Aug 01 2001
E.g.f.: exp(x)*(x+15*x^2+25*x^3+10*x^4+x^5). - Geoffrey Critzer, Jun 12 2013
a(n) = 5*a(n-1) - 10* a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + 120. - Ant King, Sep 23 2013
a(n) = n + Sum_{j=0..n-1}{k=1..4}binomial(5,k)*j^(5-k). - Patrick J. McNab, Mar 28 2016
From Kolosov Petro, Oct 22 2018: (Start)
a(n) = Sum_{k=1..n} A300656(n,k).
a(n) = Sum_{k=0..n-1} A300656(n,k). (End)
a(n) = Sum_{k=1..5} Eulerian(5, k)*binomial(n+5-k, 5), with Eulerian(5, k) = A008292(5, k), the numbers 1, 26, 66, 26, 1, for n >= 0. Worpitzki's identity for powers of 5. See. e.g., Graham et al., eq. (6, 37) (using A173018, the row reversed version of A123125). - Wolfdieter Lang, Jul 17 2019
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(5) (A013663).
Sum_{n>=1} (-1)^(n+1)/a(n) = 15*zeta(5)/16 (A267316). (End)

More terms from Henry Bottomley, Jun 21 2001

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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).

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

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

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

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

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

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

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

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)

More terms from James Sellers, Sep 19 2000

A001017 Ninth powers: a(n) = n^9.

Original entry on oeis.org

0, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, 2357947691, 5159780352, 10604499373, 20661046784, 38443359375, 68719476736, 118587876497, 198359290368, 322687697779, 512000000000, 794280046581, 1207269217792

Offset: 0

N. J. A. Sloane

A number of the form a(n) + a(n+1) + ... + a(n+k) is never prime for all n, k>=0. It could be proved by the method indicated in the comment in A256581. - Vladimir Shevelev and Peter J. C. Moses, Apr 04 2015
A generalization. Using modified Lengyel's 2007 ideas one can prove that, for every odd r>=3, every number of the form n^r + (n+1)^r + ... + (n+k)^r is nonprime. - Vladimir Shevelev, Apr 04 2015
Composition of the cubes with themselves. - Wesley Ivan Hurt, Apr 01 2016

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).

Michael De Vlieger, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
T. Lengyel, On divisibility of some power sums, INTEGERS, 7(2007), A41, 1-6.
K. MacMillan and J. Sondow, Divisibility of power sums and the generalized Erdős-Moser equation, arXiv:1010.2275 [math.NT], 2010-2011.
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Cf. A000578 (cubes), A013667 (zeta(9)), A256581.

Cf. A003391 - A004801 (sums of 2, ..., 12 positive 9th powers).

[n^9 : n in [0..40]]; // Wesley Ivan Hurt, Apr 01 2016

A001017:=n->n^9: seq(A001017(n), n=0..30); # Wesley Ivan Hurt, Apr 01 2016

Table[n^9, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
Range[0, 30]^9 (* Wesley Ivan Hurt, Apr 01 2016 *)

vector(100, n, (n-1)^9) \\ Derek Orr, Aug 03 2014

A001017(n)=n^9 \\ M. F. Hasler, Jul 03 2025

A001017 = lambda n: n**9 # M. F. Hasler, Jul 03 2025

Multiplicative with a(p^e) = p^(9e). - David W. Wilson, Aug 01 2001
Totally multiplicative sequence with a(p) = p^9 for primes p. - Jaroslav Krizek, Nov 01 2009
G.f.:  x*(1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(x-1)^10. - R. J. Mathar, Jan 07 2011
a(n) = A000578(n)^3. - Wesley Ivan Hurt, Apr 01 2016
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(9) (A013667).
Sum_{n>=1} (-1)^(n+1)/a(n) = 255*zeta(9)/256. (End)

More terms from James Sellers, Sep 19 2000

A064538 a(n) is the smallest positive integer such that a(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients.

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 42, 24, 90, 20, 66, 24, 2730, 420, 90, 48, 510, 180, 3990, 840, 6930, 660, 690, 720, 13650, 1092, 378, 56, 870, 60, 14322, 7392, 117810, 7140, 210, 72, 1919190, 103740, 8190, 1680, 94710, 13860, 99330, 9240, 217350, 9660, 9870, 10080, 324870

Offset: 0

Floor van Lamoen, Oct 08 2001

a(n) is a multiple of n+1. - Vladimir Shevelev, Dec 20 2011
Let P_n(x) = 1^n + 2^n + ... + x^n = Sum_{i=1..n+1}c_i*x^i. Let P^*n(x) = Sum{i=1..n+1}(c_i/(i+1))*(x^(i+1)-x). Then b(n) = (n+1)*a(n+1)is the smallest positive integer such that b(n)*P^*n(x) is a polynomial with integer coefficients. Proof follows from the recursion P(n+1)(x) = x + (n+1)*P^*n(x). As a corollary, note that, if p is the maximal prime divisor of a(n), then p<=n+1. - _Vladimir Shevelev, Dec 21 2011
The recursion P_(n+1)(x) = x + (n+1)*P^*n(x) is due to Abramovich (1973); see also Shevelev (2007). - _Jonathan Sondow, Nov 16 2015
The sum S_m(n) = Sum_{k=0..n} k^m can be written as S_m(n) = n(n+1)(2n+1)P_m(n)/a(m) for even m>1, or S_m(n) = n^2*(n+1)^2*P_m(n)/a(m) for odd m>1, where a(m) is the LCM of the denominators of the coefficients of the polynomial P_m/a(m), i.e., the smallest integer such that P_m defined in this way has integer coefficients. (Cf. Michon link.) - M. F. Hasler, Mar 10 2013
a(n)/(n+1) is squarefree, by Faulhaber's formula and the von Staudt-Clausen theorem on the denominators of Bernoulli numbers. - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
a(n) equals n+1 times the product of the primes p <= (n+2)/(2+(n mod 2)) such that the sum of the base-p digits of n+1 is at least p. - Bernd C. Kellner and Jonathan Sondow, May 24 2017

			1^3 + 2^3 + ... + x^3 = (x(x+1))^2/4 so a(3)=4.
1^4 + 2^4 + ... + x^4 = x(x+1)(2x+1)(3x^2+3x-1)/30, therefore a(4)=30.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprints), p. 804, Eq. 23.1.4.

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (n = 0..1000 from T. D. Noe)
V. S. Abramovich, Power sums of natural numbers, Kvant 5 (1973), 22-25. (in Russian)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, J. Integer Seq. 27 (2024), Article 24.2.8, 11 pp.; arXiv:2310.01325 [math.NT], 2023.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Meng Fai Lim, On the p-divisibility of even K-groups of the ring of integers of a cyclotomic field, arXiv:2308.04099 [math.NT], 2023.
Dr. Math, Summing n^k.
R. Mestrovic, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012.
R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
G. Michon, Faulhaber's Formula on NUMERICANA.com.
E. S. Rowland, Sums of Consecutive Powers
Vladimir Shevelev, A Short Proof of a Known Relation for Consecutive Power Sums, arXiv:0711.3692 [math.CA], 2007.
Eric Weisstein's World of Mathematics, Power Sum
Wikipedia, Faulhaber's Formula.

Cf. A144845, A195441, A256581, A286516, A286762, A286763, A324369, A324370, A324371.

A064538 := n -> denom((bernoulli(n+1,x)-bernoulli(n+1))/(n+1)): # Peter Luschny, Aug 19 2011
# Formula of Kellner and Sondow (2017):
a := proc(n) local s; s := (p,n) -> add(i,i=convert(n,base,p));
select(isprime,[$2..(n+2)/(2+irem(n,2))]);
(n+1)*mul(i,i=select(p->s(p,n+1)>=p,%)) end: seq(a(n), n=0..48); # Peter Luschny, May 14 2017

A064538[n_] := Denominator[ Together[ (BernoulliB[n+1, x] - BernoulliB[n+1])/(n+1)]];
Table[A064538[n], {n, 0, 44}] (* Jean-François Alcover, Feb 21 2012, after Maple *)

a(n) = {my(vp = Vec(bernpol(n+1, x)-bernfrac(n+1))/(n+1)); lcm(vector(#vp, k, denominator(vp[k])));} \\ Michel Marcus, Feb 07 2016

from _future_ import division
from sympy.ntheory.factor_ import digits, nextprime
def A064538(n):
    p, m = 2, n+1
    while p <= (n+2)//(2+ (n% 2)):
        if sum(d for d in digits(n+1,p)[1:]) >= p:
            m *= p
        p = nextprime(p)
    return m # Chai Wah Wu, Mar 07 2018

A064538 = lambda n: (n+1)*mul([p for p in (2..(n+2)//(2+n%2)) if is_prime(p) and sum((n+1).digits(base=p)) >= p])
print([A064538(n) for n in (0..48)]) # Peter Luschny, May 14 2017

a(n) = (n+1)*A195441(n). - Jonathan Sondow, Nov 12 2015
A001221(a(n)/(n+1)) = A001222(a(n)/(n+1)). - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
rad(a(n)) = A007947(a(n)) = A144845(n) = A324369(n+1) * A324370(n+1) * A324371(n+1). - Bernd C. Kellner, Oct 12 2023

A256385 Numbers n such that 2n^2+2n+1, 3n^2+6n+5, 6n^2+30n+55 are all composite.

Original entry on oeis.org

8, 10, 11, 15, 16, 20, 26, 27, 28, 31, 33, 36, 37, 40, 41, 43, 44, 45, 46, 49, 53, 54, 55, 57, 58, 59, 61, 64, 67, 68, 71, 73, 74, 75, 77, 78, 80, 83, 88, 89, 91, 92, 93, 95, 98, 101, 103, 105, 106, 107, 108, 111, 112, 113, 114, 116, 117, 118, 120, 121, 123

Offset: 1

Vladimir Shevelev, Mar 31 2015

Or numbers n such that n^2 + (n+1)^2 + ... + (n+k)^2 is composite for all k>=0.

For a generalization see comment in A256581.

Cf. A000290, A001844, A005918, A027865, A089306, A077654, A256546, A256581.

[n: n in [0..130] | not IsPrime(2*n^2+2*n+1) and not IsPrime(3*n^2+6*n+5) and not IsPrime(6*n^2+30*n+55)]; // Vincenzo Librandi, Apr 01 2015

Select[Range[2, 200], !PrimeQ[2 #^2 + 2 # + 1] && !PrimeQ[3 #^2 + 6 # + 5] && !PrimeQ[6 #^2 + 30 # + 55] &] (* Vincenzo Librandi, Apr 01 2015 *)
Select[Range[200],AllTrue[{2#^2+2#+1,3#^2+6#+5,6#^2+30#+55},CompositeQ]&] (* Harvey P. Dale, Jul 15 2021 *)

More terms from Peter J. C. Moses, Mar 31 2015

A256546 Numbers n such that n^4 + (n+1)^4 + ... + (n+k)^4 is composite for every k>=0.

Original entry on oeis.org

11, 17, 18, 22, 29, 32, 35, 39, 41, 44, 46, 49, 50, 51, 53, 55, 57, 59, 60, 61, 64, 66, 69, 70, 73, 75, 76, 77, 79, 81, 86, 92, 95, 96, 101, 102, 103, 107, 112, 113, 114, 116, 117, 118, 120, 125, 131, 133, 135, 137, 138, 141, 143, 144, 147, 148, 149, 150, 151

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Apr 01 2015

nonn

Number n is in the sequence if and only if the following seven numbers are all composite:
P_1(n) = 2n^4 + 4n^3 + 6n^2 + 4n + 1,
P_2(n) = 3n^4 + 12n^3 + 30n^2 + 36n + 17,
P_3(n) = 5n^4 + 40n^3 + 180n^2 + 400n + 354,
P_4(n) = 6n^4 + 60n^3 + 330n^2 + 900n + 979,
P_5(n) = 10n^4 + 180n^3 + 1710n^2 + 8100n + 15333,
P_6(n) = 15n^4 + 420n^3 + 6090n^2 + 44100n + 127687,
P_7(n) = 30n^4 + 1740n^3 + 51330n^2 + 756900n + 4463999.
For a generalization, see comment in A256581.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000583, A256385, A256503, A089306, A077654, A256385, A256581.

[n: n in [0..2*10^2] | not IsPrime(2*n^4+4*n^3+6*n^2 +4*n+1) and not IsPrime(3*n^4+12*n^3+30*n^2+36*n+17) and not IsPrime(5*n^4+40*n^3+180*n^2+400*n+354) and not IsPrime(6*n^4+60*n^3+330*n^2+900*n+979) and not IsPrime(10*n^4+ 180*n^3+1710*n^2+8100*n+15333) and not IsPrime(15*n^4+ 420*n^3+6090*n^2+44100*n+127687) and not IsPrime(30*n^4+ 1740*n^3+51330*n^2+756900*n+4463999)]; // Vincenzo Librandi, Apr 03 2015

cp's OEIS Frontend

A000584 Fifth powers: a(n) = n^5.

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

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A001017 Ninth powers: a(n) = n^9.

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A064538 a(n) is the smallest positive integer such that a(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients.

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A256385 Numbers n such that 2n^2+2n+1, 3n^2+6n+5, 6n^2+30n+55 are all composite.

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A256546 Numbers n such that n^4 + (n+1)^4 + ... + (n+k)^4 is composite for every k>=0.

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