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%I #16 Jan 03 2024 08:07:40
%S 1,4,7,12,13,24,19,32,33,44,31,68,37,64,75,80,49,108,55,124,109,104,
%T 67,176,105,124,135,180,85,240,91,192,177,164,199,300,109,184,211,320,
%U 121,348,127,292,333,224,139,432,217,340,279,348,157,432,323,464,313,284,175,660,181
%N a(n) = Sum_{k=1..n} phi(k) * (floor(n/k)^2 - floor((n-1)/k)^2).
%F Sum_{k=1..n} a(k) = A018806(n).
%F G.f.: Sum_{k>=1} phi(k) * x^k * (1 + x^k)/(1 - x^k)^2.
%F Conjecture: a(n) = Sum_{k = 1..2*n} (-1)^k * gcd(k, 4*n). Cf. A344372. - _Peter Bala_, Jan 01 2024
%t a[n_] := Sum[EulerPhi[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^2), {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 24 2021 *)
%o (PARI) a(n) = sum(k=1, n, eulerphi(k)*((n\k)^2-((n-1)\k)^2));
%o (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))
%Y Cf. A018804, A018806, A344372, A344599, A344600.
%K nonn,easy
%O 1,2
%A _Seiichi Manyama_, May 24 2021
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