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

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, 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, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Author

Gus Wiseman, Sep 27 2024

Keywords

• nonn

Examples

			The sequence of nonsquarefree numbers (A013929) is:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, ...
with first differences (A078147):
  4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, ...
with runs:
  (4),(1),(3),(4),(2,2),(4),(1),(2),(1),(4,4,4,4),(1),(3),(1,1),(2,2,2), ...
with lengths (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, ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

For prime instead of nonsquarefree numbers we have A333254.

For run-sums instead of run-lengths we have A376264.

For squarefree instead of nonsquarefree we have A376306.

For prime-powers instead of nonsquarefree numbers we have A376309.

For compression instead of run-lengths we have A376312.

A000040 lists the prime numbers, differences A001223.

A000961 and A246655 list prime-powers, differences A057820.

A005117 lists squarefree numbers, differences A076259 (ones A375927).

A013929 lists nonsquarefree numbers, differences A078147.

Cf. A007674, A053797, A053806, A072284, A112925, A120992, A373198, A375707, A376305, A376307, A376311, A380595.

Programs

Maple

nsf:= remove(numtheory:-issqrfree, [$4..1000]):
S:= nsf[2..-1]-nsf[1..-2]:
R:= NULL: x:= 4: t:= 1:
for i from 2 to nops(S) do
  if S[i] = x then t:= t+1
  else R:= R,t; x:= S[i]; t:= 1
  fi
od:
R; # Robert Israel, Jan 27 2025

Mathematica

Length/@Split[Differences[Select[Range[1000], !SquareFreeQ[#]&]]]//Most