A344598 a(n) = Sum_{k=1..n} phi(k) * (floor(n/k)^2 - floor((n-1)/k)^2).
1, 4, 7, 12, 13, 24, 19, 32, 33, 44, 31, 68, 37, 64, 75, 80, 49, 108, 55, 124, 109, 104, 67, 176, 105, 124, 135, 180, 85, 240, 91, 192, 177, 164, 199, 300, 109, 184, 211, 320, 121, 348, 127, 292, 333, 224, 139, 432, 217, 340, 279, 348, 157, 432, 323, 464, 313, 284, 175, 660, 181
Offset: 1
Programs
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Mathematica
a[n_] := Sum[EulerPhi[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^2), {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 24 2021 *) -
PARI
a(n) = sum(k=1, n, eulerphi(k)*((n\k)^2-((n-1)\k)^2));
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PARI
my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+x^k)/(1-x^k)^2))
Formula
Sum_{k=1..n} a(k) = A018806(n).
G.f.: Sum_{k>=1} phi(k) * x^k * (1 + x^k)/(1 - x^k)^2.
Conjecture: a(n) = Sum_{k = 1..2*n} (-1)^k * gcd(k, 4*n). Cf. A344372. - Peter Bala, Jan 01 2024