A318680 a(n) = n * A318653(n).

1, 2, 9, 4, 25, 18, 49, 8, 27, 50, 121, 36, 169, 98, 225, 48, 289, 54, 361, 100, 441, 242, 529, 72, -125, 338, 405, 196, 841, 450, 961, 96, 1089, 578, 1225, 108, 1369, 722, 1521, 200, 1681, 882, 1849, 484, 675, 1058, 2209, 432, -1029, -250, 2601, 676, 2809, 810, 3025, 392, 3249, 1682, 3481, 900, 3721, 1922, 1323, 320

Offset: 1

Author

Antti Karttunen, Sep 02 2018

Keywords

• sign
• mult

Comments

Dirichlet convolution of a(n)/A299150(n) with itself gives A064549 [= n * Product_{primes p|n} p], like gives also the self-convolution of A318511(n)/A318512(n), as it is the same ratio reduced to its lowest terms. However, in contrast to A318511, this sequence is multiplicative as both A000027 and A318653 are multiplicative sequences (also, because A064549 and A299150 are both multiplicative).

A007814 gives the 2-adic valuation of this sequence, because there are no even terms in A318653.

Links

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

Crossrefs

Cf. A007947, A064549, A299150, A318511, A318512, A318653, A318681.

Programs

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_] := n * Numerator [f[n]]; Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

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&&dA007947(n)));
A318653(n) = numerator(v318653_aux[n]);
A318680(n) = (n*A318653(n));

Formula

a(n) = n * A318653(n).

a(n)/A299150(n) = A318511(n)/A318512(n).