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

1,4

Links

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

Formula

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

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[1<d<n, f[d] f[n/d], 0], {d, Divisors[n]}])];
a[n_] := Numerator[f[n]];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

Prog

(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&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
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]);

Crossrefs

Cf. A000010, A303757, A046644 (denominators).

Cf. also A317830, A317833, A317834, A317937.

Sequence in context: A117182 A143301 A225459 * A181722 A317833 A021587

Adjacent sequences: A317844 A317845 A317846 * A317848 A317849 A317850

Keyword

sign

Author

Antti Karttunen, Aug 14 2018

Status

approved