cp's OEIS Frontend

A376267 Run-lengths of first differences (A078147) of nonsquarefree numbers (A013929).

%I A376267 #17 Jan 28 2025 01:50:03
%S A376267 1,1,1,1,2,1,1,1,1,4,1,1,2,3,1,1,1,2,1,1,1,1,1,1,2,1,1,2,4,1,1,1,1,2,
%T A376267 1,1,1,1,2,1,1,2,1,1,1,2,1,1,1,1,1,1,3,1,1,1,2,1,1,1,4,1,1,1,2,1,1,3,
%U A376267 1,2,2,1,1,1,2,1,1,1,1,2,1,1,2,1,1,1,2
%N A376267 Run-lengths of first differences (A078147) of nonsquarefree numbers (A013929).
%H A376267 Robert Israel, <a href="/A376267/b376267.txt">Table of n, a(n) for n = 1..10000</a>
%e A376267 The sequence of nonsquarefree numbers (A013929) is:
%e A376267   4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, ...
%e A376267 with first differences (A078147):
%e A376267   4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, ...
%e A376267 with runs:
%e A376267   (4),(1),(3),(4),(2,2),(4),(1),(2),(1),(4,4,4,4),(1),(3),(1,1),(2,2,2), ...
%e A376267 with lengths (A376267):
%e A376267   1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, ...
%p A376267 nsf:= remove(numtheory:-issqrfree, [$4..1000]):
%p A376267 S:= nsf[2..-1]-nsf[1..-2]:
%p A376267 R:= NULL: x:= 4: t:= 1:
%p A376267 for i from 2 to nops(S) do
%p A376267   if S[i] = x then t:= t+1
%p A376267   else R:= R,t; x:= S[i]; t:= 1
%p A376267   fi
%p A376267 od:
%p A376267 R; # _Robert Israel_, Jan 27 2025
%t A376267 Length/@Split[Differences[Select[Range[1000], !SquareFreeQ[#]&]]]//Most
%Y A376267 For prime instead of nonsquarefree numbers we have A333254.
%Y A376267 For run-sums instead of run-lengths we have A376264.
%Y A376267 For squarefree instead of nonsquarefree we have A376306.
%Y A376267 For prime-powers instead of nonsquarefree numbers we have A376309.
%Y A376267 For compression instead of run-lengths we have A376312.
%Y A376267 A000040 lists the prime numbers, differences A001223.
%Y A376267 A000961 and A246655 list prime-powers, differences A057820.
%Y A376267 A005117 lists squarefree numbers, differences A076259 (ones A375927).
%Y A376267 A013929 lists nonsquarefree numbers, differences A078147.
%Y A376267 Cf. A007674, A053797, A053806, A072284, A112925, A120992, A373198, A375707, A376305, A376307, A376311, A380595.
%K A376267 nonn
%O A376267 1,5
%A A376267 _Gus Wiseman_, Sep 27 2024