A036382 Odd split numbers: have a nontrivial factorization n=ab with a and b coprime, so that L(a) + L(b) <= L(n), where L(x) = A029837(x) = ceiling(log_2(x)).

21, 33, 35, 39, 65, 69, 75, 77, 87, 91, 93, 105, 129, 133, 135, 141, 143, 145, 147, 155, 159, 161, 165, 175, 177, 183, 189, 195, 203, 217, 259, 261, 265, 267, 273, 275, 279, 285, 287, 291, 295, 297, 299, 301, 303, 305, 309, 315, 319, 321, 325, 327, 329, 339

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

All even numbers are split numbers, except that prime powers -- here powers of 2 -- are by definition excluded.

The gaps g(n) = a(n+1) - a(n) are growing up to some local maximum before suddenly dropping down to a very small value and starting a new cycle of growth. The local maxima, distinctly seen as kinks in the graph, are g(1) = 12, g(4) = 26, g(12) = 24, g(30) = 42, g(70) = 48, g(157) = 110, g(348) = 96, g(748) = 160, g(1603) = 192, g(3379) = 446, g(7076) = 384, ... They occur at indices slightly larger than twice the preceding one; every other is of size 6*2^k, k = 1,2,3,... while those in between don't seem to follow a simple pattern and are sometimes larger than the subsequent gap of size 6*2^k. - M. F. Hasler, Apr 15 2017

Examples

			s = 39 is a split number since s = 39 = 3*13, gcd(3,13)=1 and L(3) + L(13) = 2 + 4 = L(39).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A029827, A029837, A036378-A036390.

Programs

Mathematica

Select[Range[1, 340, 2], Function[n, Total@ Boole@ Map[And[Total@ Ceiling@ Log2@ # <= Ceiling@ Log2@ n, CoprimeQ @@ #] &, Map[{#, n/#} &, Rest@ Take[#, Ceiling[Length[#]/2]]]] > 0 &@ Divisors@ n]] (* Michael De Vlieger, Mar 03 2017 *)

Extensions

Name corrected by Michael De Vlieger, Mar 03 2017