A327123 Expansion of Sum_{k>=1} phi(k) * x^k / (1 + x^(2*k)), where phi = A000010.

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, 5, 29, 16, 9, 17, 25, 10, 37, 17, 13, 20, 41, 5, 41, 18, 25, 21, 45, 8, 37, 25, 17, 26, 53, 13, 45, 20, 17, 29, 57, 10, 61, 29, 25, 32, 65, 9, 65, 34, 21

Offset: 1

Author

Ilya Gutkovskiy, Sep 14 2019

Keywords

• nonn
• easy
• mult

Comments

Moebius transform of A050469.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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

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

Formula

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

From Amiram Eldar, Aug 28 2023: (Start)

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.

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)

Conjecture: a(n) = Sum_{k=1..n} sin(GCD(k,n) * Pi/2). - Velin Yanev and Vaclav Kotesovec, Jun 01 2024