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A376678
Position of first zero in the n-th differences of the primes, or 0 if it does not appear.
10
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
0,3
COMMENTS
Do the k-th differences of the primes contain a zero for all k > 1?
FORMULA
a(n) = A000720(A349643(n)) for n >= 2. - Pontus von Brömssen, Oct 17 2024
EXAMPLE
The third differences of the primes begin:
-1, 2, -4, 4, -4, 4, 0, -6, 8, ...
so a(3) = 7.
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]]]}]
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.
Sequence in context: A322223 A173226 A094223 * A217069 A323673 A366359
KEYWORD
nonn,changed
AUTHOR
Gus Wiseman, Oct 14 2024
EXTENSIONS
a(17)-a(32) from Pontus von Brömssen, Oct 17 2024
a(33)-a(35) from Lucas A. Brown, Nov 03 2024
STATUS
approved