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A342000
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a(1) = 0, and for n > 1, a(n) = 1 if G(A329886(n)) >= G(A329886(floor(n/2))), otherwise 0, where G(n) = sigma(n) / (n*log(log(n))), where sigma is the sum of the divisors.
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6
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0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0
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OFFSET
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1
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COMMENTS
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Equivalently, for n > 3, a(n) = 1 if the value of H(A329886(n)) <= H(A329886(n\2)), where H(n) = log(n)^(n/sigma(n)) [= exp(1/G(n))], otherwise 0.
The ratio G(n) = sigma(n) / (n*log(log(n))) comes from Grönwall's theorem (listed, for example, as Theorem 1 in the Caveney, Nicolas and Sondow paper; see also the other papers linked at A073751). This sequence gives the positions of those points in the A329886-tree (Primorial inflation of Doudna-tree) where this ratio doesn't decrease when going downwards (which it mostly does, but see also A342455 for a counterexample). There are 11355 such increasing cases among the first 65536 terms, and 113134 among the first 2^20, although overall the ratio of such cases on each row k (that has 2^k terms) seems to start decreasing after the seventh row. See A342020.
It seems that the tree has infinitely long leftward branches that contain either only zeros or only ones after a while: The leftmost edge (that are primorials, A002110, in A329886) appears to consist of zeros only after its single 1 at the second term. This depends on the (so far conjectural) observation that (log(A002110(n)) + log(prime(1+n)))^(prime(1+n)) > log(A002110(n))^(1+prime(1+n)) for all n >= 1.
On the other hand, the leftward branch starting from the left child of the 496th term of the tree appears to contain only ones (corresponding to A342455 from its fifth term onward, see comments there).
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LINKS
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EXAMPLE
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The binary tree begins as:
0
..................../ \.................
1 1
0......./ \........ 0 0 ......./ \........0
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
0 1 0 0 0 0 0 0
/\ /\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \ / \
0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
with 1 marked at each node of A329886 (Primorial inflation of Doudna-tree) if its G(n) = sigma(n) / (n*log(log(n))) ratio is greater than or equal to the corresponding ratio of its parent, and 0 otherwise.
For example, A329886(1) = 2, and A329886(3) = 4. The latter has ratio G(4) = 5.357674..., while for the former (the parent of 4), the ratio is G(2) = -4.0926..., which is less than 5.357674..., therefore a(3) = 1.
For 130 = 2*65, we have A329886(65) = 60060, A329886(130) = 3063060 = A283980(60060). Here G(3063060) = 1.594960... > 1.56762... = G(60060), in other words, here again the child has a larger G-ratio than its parent, and this is the first case where it is a child obtained with A283980 instead of doubling, thus n=130 is also the first such even number after 2 for which a(n) = 1.
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MATHEMATICA
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Block[{a, b, c, f, nn = 105}, b[0] = c[1] = 1; f[n_] := DivisorSigma[1, n]/(n Log[Log[n]]); Do[b[i] = Prime[1 + BitLength[i] - DigitCount[i, 2, 1]]*b[i - 2^Floor@ Log2@ i]; c[i + 1] = Apply[Times, Flatten@ MapIndexed[ConstantArray[Prime[First[#2]], #1] &, Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &@ Sort[Flatten[ConstantArray[PrimePi@ #1, #2] & @@@ FactorInteger[b[i]]], Greater]]]; a[i - 1] = Boole[f[c[i]] >= f[c[Floor[(i + 1)/2]]]], {i, nn}]; Array[a, nn - 1]] (* Michael De Vlieger, Mar 07 2021 *)
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PROG
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(PARI)
default(realprecision, 10001);
G(n) = (sigma(n) / (n*log(log(n))));
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
(PARI)
\\ Alternative program, probably less vulnerable to the loss of precision:
Gie(n) = (log(n)^(n/sigma(n))); \\ = exp(1/G(n)), function H in comments
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CROSSREFS
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Cf. A000203, A002110, A003961, A004394, A004490, A005940, A025487, A057641, A058209, A073751, A108951, A329886, A342012, A342013, A342455.
Cf. also A336834 for a similarly constructed sequence.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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