A389978 Arithmetic mean of A367866(n), A389977(n), A367865(n), and A389775(n).

1, 4, 9, 4, 25, 43, 49, 4, 9, 111, 121, 43, 169, 211, 241, 4, 289, 43, 361, 111, 463, 507, 529, 43, 25, 703, 9, 211, 841, 1238, 961, 4, 1123, 1191, 1261, 43, 1369, 1483, 1561, 111, 1681, 2330, 1849, 507, 241, 2163, 2209, 43, 49, 111, 2653, 703, 2809, 43, 3081, 211, 3307, 3423, 3481, 1238

Offset

1,2

Comments

The sum A367866(n) + A389977(n) + A367865(n) + A389775(n) is divisible by 4 for all n, so a(n) is always an integer.

a(n) is not multiplicative, however all of the involved sequences are.

A367866, A389977, A367865, and A389775 are the only four arithmetic functions multiplicative by a monic degree 2 Littlewood polynomial of p.

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)).

a(n) = (A367866(n) + A389977(n) + A367865(n) + A389775(n)) / 4.

a(n) = (1/4) * Sum_{d|n} d*mu(d)*((-1)^omega(n) + mu(d))*(sigma(d) + phi(d)).

Sum_{k=1..n} a(k) ~ c * n^3 / 12, where c = 1 + zeta(3) * Product_{p prime} (1 - 3/p^3 + 2/p^4) + Product_{p prime} (1 - 2/(1+p+p^2)) + Product_{p prime} (1 - 2/p + (2*p)/(1+p+p^2)) = 2.68903090563510601643... . - Amiram Eldar, Oct 21 2025

a(p) = p^2 for prime p.

a(p1*p2) = (p1*p2)^2 + p1*p2 + 1 for primes p1,p2.

a(p1*p2*p3) = (p1*p2*p3)^2 + p1^2*p2*p3 + p1*p2^2*p3 + p1*p2*p3^2 + p1 + p2 + p3 for primes p1,p2,p3.

Example

a(2) = (A367866(2) + A389977(2) + A367865(2) + A389775(2)) / 4 = (7 + 5 + 3 + 1) / 4 = 4

Mathematica

f1[p_, e_] := p^2 + p + 1; f2[p_, e_] := p^2 + p - 1; f3[p_, e_] := p^2 - p + 1; f4[p_, e_] := p^2 - p - 1; a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, (Times @@ f1 @@@ fct + Times @@ f2 @@@ fct + Times @@ f3 @@@ fct + Times @@ f4 @@@ fct) / 4]; Array[a, 60] (* Amiram Eldar, Oct 21 2025 *)

Prog

(PARI) a(n)=sumdiv(n, d, d*moebius(d)*((-1)^omega(n)+moebius(d))*(sigma(d)+eulerphi(d)))>>2;
(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, 2);
(Python)
from math import prod
from itertools import product
from sympy import primefactors
def A389978(n):
    pf = primefactors(n)
    return sum(prod(p*(p+a)+b for p in pf) for a, b in product((-1, 1), repeat=2))>>2 # Chai Wah Wu, Oct 27 2025

Crossrefs

Cf. A001221 (omega), A008683 (mu), A008966 (mu^2), A000203 (sigma), A000010 (phi), A007947 (rad).

Cf. A065387, A023900, A055615, A076479.

Averaged functions: A367866, A367865, A389775, A389978.

Sequence in context: A087321 A053143 A068238 * A280441 A255290 A087369

Adjacent sequences: A389975 A389976 A389977 * A389979 A389980 A389981

Keyword

nonn,easy

Author

Aloe Poliszuk, Oct 20 2025

Status

approved