A377042 Position of first zero in the n-th differences of the squarefree numbers (A005117), or 0 if it does not appear.

0, 0, 1, 11, 8, 57, 14, 11, 13, 1019, 44, 1250, 43, 2721, 42, 249522, 2840, 1989839, 2839, 3373774, 4933, 142715511, 42793, 435650856, 5266, 30119361, 104063, 454172978707, 100285, 434562125244, 2755089, 2409925829164, 2485612

Offset: 0

Author

Gus Wiseman, Oct 18 2024

Keywords

• nonn
• more
• hard

Comments

a(n) for n even appear to be smaller than a(n) for n odd. - Chai Wah Wu, Oct 19 2024

a(33) > 10^13, unless it is 0. - Lucas A. Brown, Nov 15 2024

Examples

			The fourth differences begin: -3, 3, 1, -6, 7, -5, 3, 0, -2, ... so a(4) = 8
		

Links

• Lucas A. Brown, Python program.

Crossrefs

The version for primes is A376678, noncomposites A376855, composites A377037.

This is the first position of 0 in each row of A377038.

For nonsquarefree numbers we have A377050.

For prime-powers we have A377055.

A000040 lists the primes, differences A001223, seconds A036263.

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

A073576 counts integer partitions into squarefree numbers, factorizations A050320.

A377039 gives antidiagonal-sums of A377038, absolute version A377040.

A377041 gives first column of A377038, for primes A007442 or A030016.

Cf. A007674, A053797, A053806, A061398, A072284, A076259, A112925, A120992, A376311, A376590, A376591, A377046.

Programs

Mathematica

nn=10000;
u=Table[Differences[Select[Range[nn],SquareFreeQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]

Extensions

a(15)-a(20) from Chai Wah Wu, Oct 19 2024

a(21)-a(32) from Lucas A. Brown, Nov 15 2024