A317847 Numerators of sequence whose Dirichlet convolution with itself yields A303757, the ordinal transform of function a(1) = 0; a(n) = phi(n) for n > 1, where phi is Euler's totient function (A000010).

1, 1, 1, 7, 1, 5, 1, 9, 7, 5, 1, 15, 1, 5, 1, 43, 1, 15, 1, 7, 3, 3, 1, 5, 3, 5, 9, 15, 1, 9, 1, 87, 3, 5, 1, 1, 1, 5, 3, 13, 1, 11, 1, 11, 15, 3, 1, 187, 7, 19, 1, 15, 1, 5, 3, 21, 3, 3, 1, -1, 1, 3, 11, 387, 1, 9, 1, 7, 1, 13, 1, 119, 1, 7, 19, 23, 3, 19, 1, 139, -21, 7, 1, 21, 1, 5, 1, 39, 1, 67, 3, 3, 5, 3, 5, 451, 1, 15, 19, 69, 1, 13, 1, -27, 7

Offset: 1

Author

Antti Karttunen, Aug 14 2018

Keywords

• sign
• frac

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000010, A303757, A046644 (denominators).

Cf. also A317830, A317833, A317834, A317937.

Programs

Mathematica

A303757[n_] := If[n == 2, 1, Count[EulerPhi[Range[n]] - EulerPhi[n], 0]];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A303757[n] -
     Sum[If[1Jean-François Alcover, Dec 20 2021 *)

PARI

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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&&dA317937.
v303757 = ordinal_transform(vector(up_to,n,if(1==n,0,eulerphi(n))));
v317847 = DirSqrt(vector(up_to, n, v303757[n]));
A317847(n) = numerator(v317847[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A303757(n) - Sum_{d|n, d>1, d 1.