A375736 - OEIS

%I #7 Sep 11 2024 10:07:16

%S 1,2,1,2,1,1,1,1,2,1,1,1,1,1,1,3,1,1,2,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,

%T 1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1

%N Length of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

%C Non-perfect-powers (A007916) are numbers with no proper integer roots.

%C An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.

%e The initial anti-runs are the following, whose lengths are a(n):

%e   (2)

%e   (3,5)

%e   (6)

%e   (7,10)

%e   (11)

%e   (12)

%e   (13)

%e   (14)

%e   (15,17)

%e   (18)

%e   (19)

%e   (20)

%e   (21)

%e   (22)

%e   (23)

%e   (24,26,28)

%t radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;

%t Length/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most

%Y For squarefree numbers we have A373127, runs A120992.

%Y For nonprime numbers we have A373403, runs A176246.

%Y For nonsquarefree numbers we have A373409, runs A053797.

%Y For prime-powers we have A373576, runs A373675.

%Y For non-prime-powers (exclusive) we have A373672, runs A110969.

%Y For runs instead of anti-runs we have A375702.

%Y For anti-runs of non-perfect-powers:

%Y - length: A375736 (this)

%Y - first: A375738

%Y - last: A375739

%Y - sum: A375737

%Y For runs of non-perfect-powers:

%Y - length: A375702

%Y - first: A375703

%Y - last: A375704

%Y - sum: A375705

%Y A001597 lists perfect-powers, differences A053289.

%Y A007916 lists non-perfect-powers, differences A375706.

%Y Cf. A007674, A045542, A046933, A216765, A251092, A373679, A375714.

%K nonn

%O 1,2

%A _Gus Wiseman_, Sep 10 2024