A349236 Gaps 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
Offset: 1
Keywords
Examples
a(1) = A004709(2) - A004709(1) = 2 - 1 = 1. a(7) = A004709(8) - A004709(7) = 9 - 7 = 2.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, The distribution of k-free numbers, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; arXiv preprint, arXiv:1912.04972 [math.NT], 2019-2020.
Programs
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Mathematica
cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; Differences @ Select[Range[100], cubeFreeQ]
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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
Formula
Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = zeta(3) (A002117).
Comments