OFFSET
1,2
COMMENTS
The eight monic degree 3 Littlewood polynomials are x^3+x^2+x+1, x^3+x^2+x-1, x^3+x^2-x+1, x^3+x^2-x-1, x^3-x^2+x+1, x^3-x^2+x-1, x^3-x^2-x+1, and x^3-x^2-x-1.
The sum of these functions is always divisible by 8, so a(n) is always an integer.
a(n) is not multiplicative, however all of the involved functions are.
Where a(n) sums over degree 3 polynomials, A389978(n) sums over degree 2.
LINKS
Aloe Poliszuk, Table of n, a(n) for n = 1..10000
Wikipedia, Littlewood polynomial.
FORMULA
Since the multiplicity of all involved sequences does not involve the exponent e, a(n) = a(rad(n)), where rad = A007947.
a(p) = p^3 for prime p.
a(p1*p2) = (p1*p2)^3 + (p1*p2)^2 + p1*p2 + 1 for primes p1,p2.
Sum_{k=1..n} a(k) ~ c * n^4, where c = (1 + zeta(4) * (Product_{p prime} (1 - 2/p^2 - 1/p^4 + 2/p^5) + Product_{p prime} (1 - 2/p^2 + 1/p^4) + Product_{p prime} (1 - 2/p^2 + 2/p^3 - 3/p^4 + 2/p^5) + Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4) + Product_{p prime} (1 - 2/p^3 - 1/p^4 + 2/p^5) + Product_{p prime} (1 - 2/p^3 + 1/p^4) + Product_{p prime} (1 - 3/p^4 + 2/p^5))) / 32 = 0.16245695613418854481... . - Amiram Eldar, Nov 04 2025
EXAMPLE
a(2) = (15 + 13 + 11 + 9 + 7 + 5 + 3 + 1) / 8 = 8.
MATHEMATICA
f[p_, e_] := (Prepend[#, 1].{p^3, p^2, p, 1}) & /@ Tuples[{-1, 1}, 3]; a[1] = 1; a[n_] := Total[Times @@ f @@@ FactorInteger[n]] / 8; Array[a, 50] (* Amiram Eldar, Nov 04 2025 *)
PROG
(PARI)
part(S, ind, k) = prod(X=1, #S, S[X]^k + sum(Y=0, k-1, (-1)^(ind\(2^Y))*S[X]^Y));
seq(n, m) = my(fac=factorint(n)); sum(Z=1, 2^m, part(fac[, 1], Z, m))/(2^m);
a(n) = seq(n, 3);
(Python)
from itertools import product
from math import prod
from sympy import primefactors
def A386704(n):
pf = primefactors(n)
return sum(prod(p*(p*(p+a)+b)+c for p in pf) for a, b, c in product((-1, 1), repeat=3))>>3 # Chai Wah Wu, Nov 09 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Aloe Poliszuk, Oct 30 2025
STATUS
approved