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Amiram Eldar, <a href="/A349236/b349236.txt">Table of n, a(n) for n = 1..10000</a>

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(PARI)

A003557(n) = (n/factorback(factorint(n)[, 1]));

isA004709(n) = issquarefree(A003557(n));

A349236list(first_n) = { my(v=vector(first_n), k=0, e=1); for(n=2, oo, if(isA004709(n), k++; v[k] = n-e; e = n); if(#v==k, return(v))); }; \\ Antti Karttunen, Nov 11 2021

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allocated for Amiram EldarGaps between cubefree numbers: a(n) = A004709(n+1) - A004709(n).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1

1,7

This sequence is unbounded since by the Chinese Remainder Theorem there are arbitrarily long runs of consecutive numbers that are not cubefree.

The first occurrence of a(n) = 1, 2, ... is at n = 1, 7, 68, 1145, 18825, 15003967, ...

The asymptotic density of the occurrences of 1 in this sequence is density(A340152)/density(A004709) = A340153/A088453 = 0.8136635872...

Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, <a href="https://doi.org/10.1090/mcom/3581">The distribution of k-free numbers</a>, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; <a href="https://arxiv.org/abs/1912.04972">arXiv preprint</a>, arXiv:1912.04972 [math.NT], 2019-2020.

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = zeta(3) (A002117).

a(1) = A004709(2) - A004709(1) = 2 - 1 = 1.

a(7) = A004709(8) - A004709(7) = 9 - 7 = 2.

cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; Differences @ Select[Range[100], cubeFreeQ]

Cf. A002117, A046099, A004709, A076259, A088453, A271443, A340152, A340153.

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Amiram Eldar, Nov 11 2021

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