A376678 - OEIS

%I #21 Nov 04 2024 09:30:31

%S 0,0,2,7,69,13,47,58,9,43,3553,100,7019,14082,68097,14526,149677,2697,

%T 481054,979719,631894,29811,25340978,50574254,7510843,210829337,

%U 67248861,224076286,910615647,931510269,452499644,2880203722,396680865,57954439970,77572822440,35394938648

%N Position of first zero in the n-th differences of the primes, or 0 if it does not appear.

%C Do the k-th differences of the primes contain a zero for all k > 1?

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A376678.py">Python program</a>.

%F a(n) = A000720(A349643(n)) for n >= 2. - _Pontus von Brömssen_, Oct 17 2024

%e The third differences of the primes begin:

%e   -1, 2, -4, 4, -4, 4, 0, -6, 8, ...

%e so a(3) = 7.

%t nn=100000;

%t u=Table[Differences[Select[Range[nn],PrimeQ],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 If 1 is considered prime (A008578) we get A376855.

%Y The zeros of second differences are A064113, complement A333214.

%Y This is the position at which 0 first appears in row n of A095195.

%Y For composite instead of prime we have A377037.

%Y For squarefree instead of prime we have A377042, nonsquarefree A377050.

%Y For prime-power instead of prime we have A377055.

%Y A000040 lists the primes, first differences A001223, second A036263.

%Y Cf. A000720, A007442, A030016, A065890, A084758, A140119, A258025, A258026, A333254, A349643, A376681, A376682, A376683.

%K nonn

%O 0,3

%A _Gus Wiseman_, Oct 14 2024

%E a(17)-a(32) from _Pontus von Brömssen_, Oct 17 2024

%E a(33)-a(35) from _Lucas A. Brown_, Nov 03 2024