A317925 Numerators of rational valued sequence whose Dirichlet convolution with itself yields Euler's phi (A000010).

1, 1, 1, 7, 2, 1, 3, 25, 5, 1, 5, 7, 6, 3, 2, 363, 8, 5, 9, 7, 3, 5, 11, 25, 8, 3, 13, 21, 14, 1, 15, 1335, 5, 4, 6, 35, 18, 9, 6, 25, 20, 3, 21, 35, 5, 11, 23, 363, 33, 4, 8, 21, 26, 13, 10, 75, 9, 7, 29, 7, 30, 15, 15, 9923, 12, 5, 33, 7, 11, 3, 35, 125, 36, 9, 8, 63, 15, 3, 39, 363, 139, 10, 41, 21, 16, 21, 14, 125, 44, 5, 18, 77, 15, 23

Offset: 1

Author

Antti Karttunen, Aug 11 2018

Keywords

• nonn
• frac

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A000010, A317926 (denominators).

Cf. also A046643, A317831.

Programs

Mathematica

f[1] = 1; f[n_] := f[n] = (EulerPhi[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; Numerator @ Array[f, 100] (* Amiram Eldar, Dec 12 2022 *)

PARI

A317925perA317926(n) = if(1==n,n,(eulerphi(n)-sumdiv(n,d,if((d>1)&&(dA317925perA317926(d)*A317925perA317926(n/d),0)))/2);
A317925(n) = numerator(A317925perA317926(n));

PARI

\\ Memoized implementation:
memo = Map();
A317925perA317926(n) = if(1==n,n,if(mapisdefined(memo,n),mapget(memo,n),my(v = (eulerphi(n)-sumdiv(n,d,if((d>1)&&(dA317925perA317926(d)*A317925perA317926(n/d),0)))/2); mapput(memo,n,v); (v)));

PARI

for(n=1, 100, print1(numerator(direuler(p=2, n, ((1-X)/(1-p*X))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A000010(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A317925(k) / A317926(k) ~ Pi^(-3/2) * n^2 * sqrt(3/(2*log(n))) * (1 + (1/2 - gamma/2 + 3*zeta'(2)/Pi^2) / (2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 10 2025