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

0, 0, 2, 7, 69, 13, 47, 58, 9, 43, 3553, 100, 7019, 14082, 68097, 14526, 149677, 2697, 481054, 979719, 631894, 29811, 25340978, 50574254, 7510843, 210829337, 67248861, 224076286, 910615647, 931510269, 452499644, 2880203722, 396680865, 57954439970, 77572822440, 35394938648

Offset: 0

Author

Gus Wiseman, Oct 14 2024

Keywords

• nonn

Comments

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

Examples

			The third differences of the primes begin:
  -1, 2, -4, 4, -4, 4, 0, -6, 8, ...
so a(3) = 7.
		

Links

• Lucas A. Brown, Python program.

Crossrefs

If 1 is considered prime (A008578) we get A376855.

The zeros of second differences are A064113, complement A333214.

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

For composite instead of prime we have A377037.

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

For prime-power instead of prime we have A377055.

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

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

Programs

Mathematica

nn=100000;
u=Table[Differences[Select[Range[nn],PrimeQ],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]]]}]

Formula

a(n) = A000720(A349643(n)) for n >= 2. - Pontus von Brömssen, Oct 17 2024

Extensions

a(17)-a(32) from Pontus von Brömssen, Oct 17 2024

a(33)-a(35) from Lucas A. Brown, Nov 03 2024