A344596 a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^3 - floor((n-1)/k)^3).
1, 6, 18, 30, 60, 66, 126, 132, 198, 204, 330, 276, 468, 414, 552, 552, 816, 630, 1026, 840, 1116, 1050, 1518, 1128, 1740, 1476, 1890, 1692, 2436, 1704, 2790, 2256, 2820, 2544, 3384, 2556, 3996, 3186, 3960, 3408, 4920, 3420, 5418, 4260, 5112, 4686, 6486, 4560, 6930, 5340, 6816
Offset: 1
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
a[n_] := Sum[MoebiusMu[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^3), {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 24 2021 *) -
PARI
a(n) = sum(k=1, n, moebius(k)*((n\k)^3-((n-1)\k)^3));
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PARI
a(n) = if(n<2, n, 3*sumdiv(n, d, moebius(n/d)*(d-1)*d));
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PARI
my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3)) -
PARI
my(N=66, x='x+O('x^N)); Vec(x+6*sum(k=1, N, moebius(k)*x^(2*k)/(1-x^k)^3)) -
Python
from sympy import mobius, divisors def A344596(n): return 3*sum(mobius(n//d)*d*(d-1) for d in divisors(n,generator=True)) if n>1 else 1 # Chai Wah Wu, May 09 2025
Formula
Sum_{k=1..n} a(k) * floor(n/k) = n^3.
Sum_{k=1..n} a(k) = A071778(n).
a(n) = 3 * Sum_{d|n} mu(n/d) * (d-1) * d for n > 1.
G.f.: Sum_{k >= 1} mu(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3.
G.f.: x + 6 * Sum_{k>=1} mu(k) * x^(2*k)/(1 - x^k)^3.
a(2^k) = 3*2^(k-2)*(3*2^k-2) for k>0. - Chai Wah Wu, May 10 2025