A300909 Sum of 4th powers dividing n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 82, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17

Offset

1,16

Comments

Multiplicative with a(p^e) = (p^(4*(1+floor(e/4)))-1)/(p^4-1). - Robert Israel, Mar 15 2018

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

A. Dixit, B. Maji, A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=4,k=4,n).

Formula

G.f.: Sum_{k>=1} k^4*x^(k^4)/(1 - x^(k^4)).

L.g.f.: -log(Product_{k>=1} (1 - x^(k^4))) = Sum_{n>=1} a(n)*x^n/n.

D.g.f.: zeta(s)*zeta(4s-4). - Robert Israel, Mar 15 2018

Sum_{k=1..n} a(k) ~ zeta(5/4)*n^(5/4)/5 - n/2. - Vaclav Kotesovec, Dec 01 2020

Example

a(16) = 17 because 16 has 5 divisors {1, 2, 4, 8, 16} among which 2 divisors {1, 16} are 4th powers and 1 + 16 = 17.
L.g.f.: L(x) = x + x^2/2 + x^3/3 + x^4/4 + x^5/5 + x^6/6 + x^7/7 + x^8/8 + x^9/9 + x^10/10 + x^11/11 + x^12/12 + x^13/13 + x^14/14 + x^15/15 + 17*x^16/16 + x^17/17 + ...
exp(L(x)) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + 2*x^16 + 2*x^17 + ... + A046042(n)*x^n + ...

Maple

N:= 1000: # for a(1)..a(N)
V:= Vector(N, 1):
for m from 2 to floor(N^(1/4)) do
  R:= [seq(i, i=m^4 .. N, m^4)];
  V[R]:= map(`+`, V[R], m^4)
od:
convert(V, list); # Robert Israel, Mar 15 2018

Mathematica

Table[DivisorSum[n, # &, IntegerQ[#^(1/4)] &], {n, 112}]
nmax = 112; Rest[CoefficientList[Series[Sum[k^4 x^k^4/(1 - x^k^4), {k, 1, 10}], {x, 0, nmax}], x]]
nmax = 112; Rest[CoefficientList[Series[-Log[Product[(1 - x^k^4), {k, 1, 10}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := (p^(4*(1 + Floor[e/4])) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

Prog

(PARI) a(n) = sumdiv(n, d, d*ispower(d, 4)); \\ Michel Marcus, Mar 15 2018

Crossrefs

Cf. A000583, A001159, A035316, A046042, A046100 (positions of ones), A063775, A113061.

Sequence in context: A335264 A241027 A176728 * A088469 A089170 A040292

Adjacent sequences: A300906 A300907 A300908 * A300910 A300911 A300912

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Mar 15 2018

Status

approved