A327123 - OEIS

%I #22 Jun 01 2024 12:06:11

%S 1,1,1,2,5,1,5,4,5,5,9,2,13,5,5,8,17,5,17,10,5,9,21,4,25,13,13,10,29,

%T 5,29,16,9,17,25,10,37,17,13,20,41,5,41,18,25,21,45,8,37,25,17,26,53,

%U 13,45,20,17,29,57,10,61,29,25,32,65,9,65,34,21

%N Expansion of Sum_{k>=1} phi(k) * x^k / (1 + x^(2*k)), where phi = A000010.

%C Moebius transform of A050469.

%H Amiram Eldar, <a href="/A327123/b327123.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} mu(n/d) * A050469(d).

%F From _Amiram Eldar_, Aug 28 2023: (Start)

%F Multiplicative with a(2^e) = 2^(e-1), and if p is an odd prime a(p^e) = 1 if p == 1 (mod 4) and (p^(e+1) - p^e + 2*(-1)^e)/(p+1) otherwise.

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*G/Pi^2 = 0.278420154533..., and G is Catalan's constant (A006752). (End)

%F Conjecture: a(n) = Sum_{k=1..n} sin(GCD(k,n) * Pi/2). - _Velin Yanev_ and _Vaclav Kotesovec_, Jun 01 2024

%t nmax = 69; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t A050469[n_] := DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 4]] &] - DivisorSum[n, # &, MemberQ[{3}, Mod[n/#, 4]] &]; a[n_] := DivisorSum[n, MoebiusMu[n/#] A050469[#] &]; Table[a[n], {n, 1, 69}]

%t f[p_, e_] := If[Mod[p, 4] == 1, p^e, (p^(e+1) - p^e + 2*(-1)^e)/(p+1)]; f[2, e_] := 2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* _Amiram Eldar_, Aug 28 2023 *)

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p == 2, 2^(e-1), if(p%4 == 1, p^e, (p^(e+1) - p^e + 2*(-1)^e)/(p+1)))); } \\ _Amiram Eldar_, Aug 28 2023

%Y Cf. A000010, A004613 (fixed points), A006752, A008683, A026741, A050469, A101455, A193356, A327122.

%K nonn,easy,mult

%O 1,4

%A _Ilya Gutkovskiy_, Sep 14 2019