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A375707 First differences minus 1 of nonsquarefree numbers.

%I A375707 #17 Sep 17 2024 04:02:39
%S A375707 3,0,2,3,1,1,3,0,1,0,3,3,3,3,0,2,0,0,1,1,1,3,2,0,3,3,2,0,3,0,2,3,1,1,
%T A375707 3,1,0,0,3,3,3,3,0,2,0,2,0,0,1,3,2,0,3,3,2,0,1,1,0,2,3,1,1,3,0,1,0,2,
%U A375707 0,3,3,3,0,2,3,1,1,3,2,0,3,3,3,3,0,2,3
%N A375707 First differences minus 1 of nonsquarefree numbers.
%C A375707 Also the number of squarefree numbers between the nonsquarefree numbers A013929(n) and A013929(n+1).
%C A375707 Delete all 0's to get A120992.
%C A375707 The image is {0,1,2,3}.
%C A375707 Add 1 to all terms for A078147.
%H A375707 Amiram Eldar, <a href="/A375707/b375707.txt">Table of n, a(n) for n = 1..10000</a>
%F A375707 Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 6/(Pi^2-6) = 1.550546... . - _Amiram Eldar_, Sep 17 2024
%e A375707 The runs of squarefree numbers begin:
%e A375707   (5,6,7)
%e A375707   ()
%e A375707   (10,11)
%e A375707   (13,14,15)
%e A375707   (17)
%e A375707   (19)
%e A375707   (21,22,23)
%e A375707   ()
%e A375707   (26)
%e A375707   ()
%e A375707   (29,30,31)
%e A375707   (33,34,35)
%t A375707 Differences[Select[Range[100],!SquareFreeQ[#]&]]-1
%o A375707 (PARI) lista(nmax) = {my(prev = 4); for (n = 5, nmax, if(!issquarefree(n), print1(n - prev - 1, ", "); prev = n));} \\ _Amiram Eldar_, Sep 17 2024
%Y A375707 Positions of 0, 1, 2, 3 are A375709, A375710, A375711, A375712. This is a set partition of the positive integers into four blocks.
%Y A375707 For runs of squarefree numbers:
%Y A375707 - length: A120992, anti A373127
%Y A375707 - min: A072284, anti A373408
%Y A375707 - max: A373415, anti A007674
%Y A375707 - sum: A373413, anti A373411
%Y A375707 For runs of nonsquarefree numbers:
%Y A375707 - length: A053797, anti A373409
%Y A375707 - min: A053806, anti A373410
%Y A375707 - max: A376164, anti A068781
%Y A375707 - sum: A373414, anti A373412
%Y A375707 A005117 lists the squarefree numbers, first differences A076259.
%Y A375707 A013929 lists the nonsquarefree numbers, first differences A078147.
%Y A375707 A046933 counts composite numbers between consecutive primes.
%Y A375707 A073784 counts primes between consecutive composite numbers.
%Y A375707 A093555 counts non-prime-powers between consecutive prime-powers.
%Y A375707 Cf. A061398, A061399, A077641, A077643, A143658, A173143, A251092, A294242, A373125, A373126, A373573.
%K A375707 nonn,easy
%O A375707 1,1
%A A375707 _Gus Wiseman_, Sep 16 2024