A376680 - OEIS

%I #6 Oct 10 2024 08:52:44

%S 2,2,2,2,2,2,1,4,2,4,1,2,2,2,1,4,1,4,2,4,1,2,2,4,1,2,1,4,1,6,1,2,2,2,

%T 2,2,1,12,1,2,1,4,2,8,2,4,1,4,1,2,1,4,1,4,2,8,2,2,2,10,1,10,1,2,2,2,1,

%U 4,2,8,1,4,1,4,1,4,2,4,1,2,2,8,1,12,1,2

%N Run-lengths of first differences of composite numbers.

%C Also first differences of A376603 (points of nonzero curvature in the composite numbers).

%e The composite numbers (A002808) are:

%e   4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...

%e with first differences (A073783):

%e   2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, ...

%e with runs:

%e   (2,2), (1,1), (2,2), (1,1), (2,2), (1,1), (2), (1,1,1,1), (2,2), (1,1,1,1), ...

%e with lengths (A376680):

%e   2, 2, 2, 2, 2, 2, 1, 4, 2, 4, 1, 2, 2, 2, 1, 4, 1, 4, 2, 4, 1, 2, 2, 4, 1, 2, ...

%t Length/@Split[Differences[Select[Range[100],CompositeQ]]]

%Y These are the run-lengths of A073783, ones A375929.

%Y For prime instead of composite we have A333254, first appearances A335406.

%Y These are the first differences of A376603.

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

%Y A002808 lists the composite numbers, differences A073783.

%Y A064113 lists positions of adjacent equal prime gaps.

%Y A073445 gives second differences of composite numbers, zeros A376602.

%Y Cf. A054546, A076259, A174965, A251092, A376651, A376652.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 10 2024