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A318653 Numerators of the sequence whose Dirichlet convolution with itself yields A007947, the squarefree kernel of n. 4
1, 1, 3, 1, 5, 3, 7, 1, 3, 5, 11, 3, 13, 7, 15, 3, 17, 3, 19, 5, 21, 11, 23, 3, -5, 13, 15, 7, 29, 15, 31, 3, 33, 17, 35, 3, 37, 19, 39, 5, 41, 21, 43, 11, 15, 23, 47, 9, -21, -5, 51, 13, 53, 15, 55, 7, 57, 29, 59, 15, 61, 31, 21, 5, 65, 33, 67, 17, 69, 35, 71, 3, 73, 37, -15, 19, 77, 39, 79, 15, 3, 41, 83, 21, 85, 43, 87, 11, 89, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

No zeros among the first 2^20 terms.

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) * (A007947(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

MATHEMATICA

rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); f[1] = 1; f[n_] := f[n] = (rad[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; a[n_] := Numerator [f[n]]; Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

PROG

(PARI)

up_to = 65537;

A007947(n) = factorback(factorint(n)[, 1]);

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};

v318653_aux = DirSqrt(vector(up_to, n, A007947(n)));

A318653(n) = numerator(v318653_aux[n]);

CROSSREFS

Cf. A007947, A299150 (denominators).

Cf. also A317935, A318511, A318512, A318649.

Sequence in context: A299766 A161398 A204455 * A161820 A341042 A116528

Adjacent sequences:  A318650 A318651 A318652 * A318654 A318655 A318656

KEYWORD

sign,frac,mult

AUTHOR

Antti Karttunen, Aug 31 2018

STATUS

approved

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Last modified June 16 10:23 EDT 2021. Contains 345056 sequences. (Running on oeis4.)