A321973 Partial sums of the Dedekind psi_2(k) function, for 1 <= k <= n.

0, 1, 6, 16, 36, 62, 112, 162, 242, 332, 462, 584, 784, 954, 1204, 1464, 1784, 2074, 2524, 2886, 3406, 3906, 4516, 5046, 5846, 6496, 7346, 8156, 9156, 9998, 11298, 12260, 13540, 14760, 16210, 17510, 19310, 20680, 22490, 24190, 26270, 27952, 30452, 32302, 34742

Offset

0,3

Comments

In general, for m >= 1, Sum_{k=1..n} psi_m(k) = Sum_{k=1..n} mu(k)^2 * (Bernoulli(m+1, 1+floor(n/k)) - Bernoulli(m+1, 1)) / (m+1), where mu(k) is the Moebius function and Bernoulli(n,x) are the Bernoulli polynomials.

In general, for m >= 1, Sum_{k=1..n} psi_m(k) ~ n^(m+1) * zeta(m+1) / ((m+1) * zeta(2*(m+1))).

Links

Table of n, a(n) for n=0..44.

Wikipedia, Dedekind psi function

Formula

a(n) = Sum_{k=1..n} A065958(k).

a(n) ~ n^3 * zeta(3) / (3*zeta(6)).

a(n) = Sum_{k=1..n} mu(k)^2 * Bernoulli(3, 1+floor(n/k)) / 3.

Mathematica

a[n_] := Sum[MoebiusMu[k]^2 * BernoulliB[3, 1 + Floor[n/k]], {k, 1, n}]/3; Array[a, 50, 0] (* Amiram Eldar, Nov 23 2018 *)

Prog

(PARI) a(n) = sum(k=1, n, moebius(k)^2 * ((n\k)^3/3 + (n\k)^2/2 + (n\k)/6));
(Python)
from math import isqrt
from functools import lru_cache
from sympy import mobius
def A321973(n):
    @lru_cache(maxsize=None)
    def f(x): return mobius(x)
    c, j, v = 0, 1, 0
    while j <= n:
        k = n//j
        m = n//k
        c += k*(k*(2*k+3)+1)*(-v+(v:=sum(f(r)*(m//r**2) for r in range(1, isqrt(m)+1))))
        j = m+1
    return c//6 # Chai Wah Wu, May 16 2026

Crossrefs

Cf. A001615, A065958, A173290.

Sequence in context: A334047 A199629 A098943 * A178465 A247619 A120586

Adjacent sequences: A321970 A321971 A321972 * A321974 A321975 A321976

Keyword

nonn,easy

Author

Daniel Suteu, Nov 22 2018

Status

approved