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A338026 a(1) = 8; for n > 1, a(n) is the largest integer m such that m = ((2*x*a(n-1)) / (x+1)) - x, with x a positive nontrivial divisor of m.

%I A338026 #32 Apr 12 2025 09:15:28
%S A338026 8,9,10,12,15,20,28,42,66,110,190,342,630,1190,2278,4422,8646,17030,
%T A338026 33670,66822,132900,264758,528034,1053990,2105077,4205820,8405840,
%U A338026 16803405,33595212,67173930,134324628,268616475,537185908,1074305622,2148516546
%N A338026 a(1) = 8; for n > 1, a(n) is the largest integer m such that m = ((2*x*a(n-1)) / (x+1)) - x, with x a positive nontrivial divisor of m.
%C A338026 There are no primes in the sequence and, excepting for a(1), no powers of 2.
%C A338026 For each n > 1 there are two integers f, g such that f*g = a(n) and f + g = 2*a(n-1) - a(n) - 1. (Empirical observation)
%C A338026 Excluding the condition that each term should be the largest one, the terms which satisfies the remaining conditions performs an irregular infinite net.
%C A338026 If the condition "be the largest term" is replaced for "be the smallest one" with the rest of conditions remaining, A209724 sequence is obtained. (This is true, at least, from a'(1) to a'(100))
%C A338026 Similar sequences are obtained with a'(1) a nonprime integer larger than 6, either a power of two or not. However, if a'(1) is not a power of two, there exist at least, one integer: a'(0) = ((a'(1)+x)*(x+1)) / (2*x) with x a positive nontrivial divisor of a'(1). Example: a'(1) = 26, a'(0) = 21, a'(-1) = 16. (As a'(-1) is a power of 2 there is not an a'(-2) term). If a'(1) = 6, a'(0) = a'(1) = a'(2) = ... = a'(k) = 6.
%H A338026 Enric Reverter i Bigas, <a href="https://ibb.co/NyDmBxX">Graphic example</a>
%e A338026 a(5) = 15 = ((2*3*12) / 4) - 3 or ((2*5*12) / 6) - 5 = 15; Also 14 = ((2*2*12) / 3) - 2, but 15 is larger.
%t A338026 w[n_] := Module[{x, p}, Max[p /. List@ToRules@Reduce[p == (2 n*x)/(x + 1) - x == x*y && x > 1 && y > 1, p, Integers]]]; n := 8; k := {n}; m = 1; While[m < 35, {AppendTo[k, w[n]], n = w[n]}; m++]; k
%Y A338026 Cf. A209724.
%K A338026 nonn
%O A338026 1,1
%A A338026 _Enric Reverter i Bigas_, Oct 07 2020