login
A318443
Numerators of the sequence whose Dirichlet convolution with itself yields A018804, Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).
3
1, 3, 5, 23, 9, 15, 13, 91, 59, 27, 21, 115, 25, 39, 45, 1451, 33, 177, 37, 207, 65, 63, 45, 455, 179, 75, 353, 299, 57, 135, 61, 5797, 105, 99, 117, 1357, 73, 111, 125, 819, 81, 195, 85, 483, 531, 135, 93, 7255, 363, 537, 165, 575, 105, 1059, 189, 1183, 185, 171, 117, 1035, 121, 183, 767, 46355, 225, 315, 133, 759, 225, 351, 141, 5369
OFFSET
1,2
COMMENTS
Because A018804 gets only odd values on primes, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.
LINKS
FORMULA
a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A018804(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
MATHEMATICA
a18804[n_] := Sum[n EulerPhi[d]/d, {d, Divisors[n]}];
f[1] = 1; f[n_] := f[n] = 1/2 (a18804[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
a[n_] := f[n] // Numerator;
Array[a, 72] (* Jean-François Alcover, Sep 13 2018 *)
PROG
(PARI)
up_to = 16384;
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v318443aux = DirSqrt(vector(up_to, n, A018804(n)));
A318443(n) = numerator(v318443aux[n]);
CROSSREFS
Cf. A018804, A046644 (denominators).
Cf. also A318444.
Sequence in context: A270637 A064187 A112686 * A272426 A270180 A270793
KEYWORD
nonn,frac,mult
AUTHOR
STATUS
approved