A376652 - OEIS

%I #13 Jan 17 2025 14:30:11

%S 2,6,10,13,19,24,28,31,36,42,47,51,56,59,64,71,75,79,82,95,98,104,114,

%T 119,124,127,132,138,148,152,163,174,178,181,187,196,201,206,212,217,

%U 221,230,243,247,250,263,268,278,281,286,293,298,303,306,311,318,321

%N Points of downward concavity in the sequence of composite numbers (A002808).

%C These are points at which the second differences (A073445) are negative.

%C Also positions of strict descents in the first differences (A073783) of composite numbers (A002808).

%H Robert Israel, <a href="/A376652/b376652.txt">Table of n, a(n) for n = 1..10000</a>

%H Gus Wiseman, <a href="/A376652/a376652.png">Points of downward concavity in the sequence of composite numbers</a>.

%e The composite numbers are (A002808):

%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, 2, ...

%e with second differences (A073445):

%e   0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...

%e with negative terms at (A376651):

%e   2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, ...

%p Comps:= remove(isprime, [seq(i,i=4..1000)]):

%p D1:= Comps[2..-1]-Comps[1..-2]:

%p D2:= D1[2..-1]-D1[1..-2]:

%p select(t -> D2[t] < 0, [$1..nops(D2)]); # _Robert Israel_, Nov 06 2024

%t Join@@Position[Sign[Differences[Select[Range[1000],CompositeQ],2]],-1]

%Y The version for A000002 is A156242, positive A022297.

%Y Partitions into composite numbers are counted by A023895, factorizations A050370.

%Y For first differences we had A065310 or A073783, ones A375929.

%Y These are the positions of negative terms in A073445, positive A376651.

%Y For prime instead of composite we have A258026, positive A258025.

%Y For zero second differences instead of negative we have A376602.

%Y For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (inflections and undulations), A376603 (nonzero curvature), A376651 (concave-up).

%Y Cf. A000961, A007916, A057820, A064113, A065890, A251092, A333254, A376604.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 06 2024