A318512 - OEIS

A318512

Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.

1, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 2, 4, 2, 1, 4, 1, 2, 1, 8, 1, 16, 1, 2, 2, 2, 1, 4, 1, 4, 4, 2, 1, 4, 1, 2, 2, 2, 1, 16, 1, 2, 1, 8, 4, 4, 1, 2, 8, 4, 1, 4, 1, 2, 2, 2, 1, 16, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 16, 1, 4, 2, 2, 1, 128, 1, 2, 2, 4, 1, 4, 1, 2, 8, 4, 1, 4, 1, 4, 1, 2, 4, 16, 4, 2, 2, 2, 1, 8

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OFFSET

1,3

COMMENTS

These are also denominators (in their lowest terms) for the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.

From Antti Karttunen, Sep 02 2018: (Start)

Proof for the above claim:

This sequence is defined as the denominator (given in the lowest terms) of rational valued function r(1) = 1, r(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d<n} r(d) * r(n/d)) for n > 1. Define sequence Ay(n) as the denominator of function s(n), with otherwise similar definition, but with A064549 in place of A000290. Let Ay(n) be the denominator of s(n), reduced also into the lowest terms. (Corresponding numerators are A318649 and A318511 respectively. Note that the denominators in both cases must always be of the form 2^k, with k >= 0).

By applying the distributive property of Dirichlet Convolution [which says that for any completely multiplicative function f, it doesn't matter whether one multiplies the result of convolution afterwards, or whether one multiplies the operands separately before convolution: f(g * g) = (fg) * (fg)], with A000027 in the role of f in both cases, one obtains a pair of equations:

A318649(n) A318681(n) n*A299149(n)

---------- = ---------- = ------------

A318512(n) A299150(n) A299150(n)

and

A318511(n) A318680(n) n*A318653(n)

---------- = ---------- = ------------

Ay(n) A299150(n) A299150(n)

where the leftmost ratios are reduced into their lowest terms.

Sequence A318656 gives the 2-adic valuation of ratio A318649(n)/A318512(n), and because there are no even terms neither in A299149 nor in A318653, it also gives the 2-adic valuation of the latter ratio. As A318511/Ay is given in the lowest terms (not both of A318511(n) and Ay(n) can be even at same n), this implies that Ay must indeed be identical to A318512, and furthermore that A318655(n) = A007814(A318649(n)) = A007814(A318511(n)).

(End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Wikipedia, Dirichlet convolution

FORMULA

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1. [Equally, one could use A064549 in place of A000290.]

a(n) = 2^A318513(n).

a(n) = A046644(n)/A318651(n).

a(2n-1) = A046644(2n-1) = A318658(2n-1), for all n >= 1.

PROG

(PARI)

up_to = 65537;

A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); };

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};

v318511_12 = DirSqrt(vector(up_to, n, A064549(n)));

A318512(n) = denominator(v318511_12[n]);

CROSSREFS