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A376678 Position of first zero in the n-th differences of the primes, or 0 if it does not appear.

%I A376678 #21 Nov 04 2024 09:30:31
%S A376678 0,0,2,7,69,13,47,58,9,43,3553,100,7019,14082,68097,14526,149677,2697,
%T A376678 481054,979719,631894,29811,25340978,50574254,7510843,210829337,
%U A376678 67248861,224076286,910615647,931510269,452499644,2880203722,396680865,57954439970,77572822440,35394938648
%N A376678 Position of first zero in the n-th differences of the primes, or 0 if it does not appear.
%C A376678 Do the k-th differences of the primes contain a zero for all k > 1?
%H A376678 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A376678.py">Python program</a>.
%F A376678 a(n) = A000720(A349643(n)) for n >= 2. - _Pontus von Brömssen_, Oct 17 2024
%e A376678 The third differences of the primes begin:
%e A376678   -1, 2, -4, 4, -4, 4, 0, -6, 8, ...
%e A376678 so a(3) = 7.
%t A376678 nn=100000;
%t A376678 u=Table[Differences[Select[Range[nn],PrimeQ],k],{k,2,16}];
%t A376678 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A376678 m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]
%Y A376678 If 1 is considered prime (A008578) we get A376855.
%Y A376678 The zeros of second differences are A064113, complement A333214.
%Y A376678 This is the position at which 0 first appears in row n of A095195.
%Y A376678 For composite instead of prime we have A377037.
%Y A376678 For squarefree instead of prime we have A377042, nonsquarefree A377050.
%Y A376678 For prime-power instead of prime we have A377055.
%Y A376678 A000040 lists the primes, first differences A001223, second A036263.
%Y A376678 Cf. A000720, A007442, A030016, A065890, A084758, A140119, A258025, A258026, A333254, A349643, A376681, A376682, A376683.
%K A376678 nonn
%O A376678 0,3
%A A376678 _Gus Wiseman_, Oct 14 2024
%E A376678 a(17)-a(32) from _Pontus von Brömssen_, Oct 17 2024
%E A376678 a(33)-a(35) from _Lucas A. Brown_, Nov 03 2024