A376264 Run-sums of first differences (A078147) of nonsquarefree numbers (A013929).

4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, 4, 2, 2, 16, 1, 3, 1, 3, 2, 2, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, 4, 1, 2, 1, 3, 1, 12, 1, 3, 4, 4, 4, 3, 1, 16, 1, 3, 4, 4, 4, 2, 3, 3, 4, 8, 1, 3, 4, 4, 3, 1, 3, 1, 8, 1, 3, 4, 1, 3, 4

Offset

1,1

Comments

Does the image include all positive integers? I have only verified this up to 8.

Links

Table of n, a(n) for n=1..86.

Example

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 sums (A376264):
  4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, ...

Mathematica

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

Crossrefs

Before taking run-sums we had A078147.

For nonprime instead of nonsquarefree numbers we have A373822.

Positions of first appearances are A376265, sorted A376266.

For run-lengths instead of run-sums we have A376267.

For squarefree instead of nonsquarefree we have A376307.

For prime-powers instead of nonsquarefree numbers we have A376310.

For compression instead of run-sums we have A376312.

A000040 lists the prime numbers, differences A001223.

A000961 and A246655 list prime-powers, differences A057820.

A003242 counts compressed compositions, ranks A333489.

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

A013929 lists nonsquarefree numbers, differences A078147.

Cf. A053797, A053806, A061398, A072284, A120992, A373197, A373413, A375707, A376305, A376306, A376311.

Sequence in context: A078147 A376312 A362331 * A058303 A240935 A090724

Adjacent sequences: A376261 A376262 A376263 * A376265 A376266 A376267

Keyword

nonn

Author

Gus Wiseman, Sep 26 2024

Status

approved