A377042 - OEIS

%I #15 Nov 15 2024 16:22:28

%S 0,0,1,11,8,57,14,11,13,1019,44,1250,43,2721,42,249522,2840,1989839,

%T 2839,3373774,4933,142715511,42793,435650856,5266,30119361,104063,

%U 454172978707,100285,434562125244,2755089,2409925829164,2485612

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

%C a(n) for n even appear to be smaller than a(n) for n odd. - _Chai Wah Wu_, Oct 19 2024

%C a(33) > 10^13, unless it is 0. - _Lucas A. Brown_, Nov 15 2024

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A377042.py">Python program</a>.

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

%t nn=10000;

%t u=Table[Differences[Select[Range[nn],SquareFreeQ],k],{k,2,16}];

%t mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];

%t m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]

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

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

%Y For nonsquarefree numbers we have A377050.

%Y For prime-powers we have A377055.

%Y A000040 lists the primes, differences A001223, seconds A036263.

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

%Y A073576 counts integer partitions into squarefree numbers, factorizations A050320.

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

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

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

%K nonn,more,hard

%O 0,4

%A _Gus Wiseman_, Oct 18 2024

%E a(15)-a(20) from _Chai Wah Wu_, Oct 19 2024

%E a(21)-a(32) from _Lucas A. Brown_, Nov 15 2024