A376655 Sorted positions of first appearances in the second differences of consecutive squarefree numbers (A005117).

1, 2, 3, 5, 6, 30, 61, 150, 514, 1025, 5153, 13390, 13391, 131964, 502651, 664312, 4387185, 5392318, 20613826

Offset: 1

Author

Gus Wiseman, Oct 07 2024

Keywords

• nonn
• more

Comments

Warning: Do not confuse with A246655 (prime-powers exclusive).

Examples

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, ...
with sorted first appearances at (A376655):
  1, 2, 3, 5, 6, 30, 61, 150, 514, 1025, 5153, 13390, 13391, ...
		

Crossrefs

For first differences we had A376311 (first appearances in A076259).

These are the sorted positions of first appearances in A376590.

For prime-powers instead of squarefree numbers we have A376653/A376654.

For primes instead of squarefree numbers we have A376656.

A000040 lists the prime numbers, differences A001223.

A005117 lists squarefree numbers, complement A013929 (differences A078147).

A073576 counts integer partitions into squarefree numbers, factorizations A050320.

For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).

For squarefree: A376591 (inflections and undulations), A376592 (nonzero curvature).

Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A251092, A373198, A376342.

Programs

Mathematica

q=Differences[Select[Range[1000],SquareFreeQ],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

Extensions

a(14)-a(19) from Chai Wah Wu, Oct 07 2024