A376603 - OEIS

%I #12 Oct 19 2024 08:34:37

%S 2,4,6,8,10,12,13,17,19,23,24,26,28,30,31,35,36,40,42,46,47,49,51,55,

%T 56,58,59,63,64,70,71,73,75,77,79,81,82,94,95,97,98,102,104,112,114,

%U 118,119,123,124,126,127,131,132,136,138,146,148,150,152,162,163

%N Points of nonzero curvature in the sequence of composite numbers (A002808).

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

%H Dominic McCarty, <a href="/A376603/b376603.txt">Table of n, a(n) for n = 1..1000</a>

%H Gus Wiseman, <a href="/A376603/a376603.png">Points of nonzero curvature in the sequence of composite numbers</a>.

%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, 2, ...

%e with first 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 nonzero terms at (A376603):

%e   2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, ...

%t Join@@Position[Sign[Differences[Select[Range[100],CompositeQ],2]],1|-1]

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

%Y These are the positions of nonzero terms in A073445.

%Y For first differences we had A073783, ones A375929, complement A065890.

%Y For prime instead of composite we have A333214.

%Y The complement is A376602.

%Y For upward concavity (instead of nonzero) we have A376651, downward A376652.

%Y For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (zeros), A376651 (concave-up), A376652 (concave-down).

%Y For nonzero curvature: A333214 (prime), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).

%Y Cf. A000961, A064113, A246655, A251092, A258025, A333254.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 05 2024