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A318512
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Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.
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13
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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
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1,3
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COMMENTS
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These are also denominators (in their lowest terms) for the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.
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:
---------- = ---------- = ------------
and
---------- = ---------- = ------------
where the leftmost ratios are reduced into their lowest terms.
(End)
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LINKS
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FORMULA
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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.]
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PROG
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(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]);
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CROSSREFS
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Cf. A000290, A064549, A046644, A299150, A318513, A318651, A318653, A318654, A318655, A318656, A318658, A318680, A318681.
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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The main definition changed, more formulas added by Antti Karttunen, Aug 31 2018
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STATUS
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approved
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