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%I #26 May 27 2014 12:28:06
%S 3,5,7,9,11,13,17,19,23,25,27,31,37,43,47,49,53,71,79,70,89,97,103,87,
%T 113,139,157,163,191,181,199,223,241,239,271,251,311,313,293,347,353,
%U 383,397,421,463,499,523,541,467,577,607,619,613,661,631,751,719
%N a(n) is the largest number that can be written in the form a(n) = 2^k1 * p1^k2 + 2^k3 * p2^k4 in n distinct ways, where p1 and p2 are odd prime numbers and k1, k2, k3, and k4 are nonnegative integers.
%C It is conjectured that, for any positive integer n, there exist only a finite number of positive integers that can be written in the form 2^k1 * p1^k2 + 2^k3 * p2^k4 in n distinct ways.
%C The Mathematica program used to generate the first 57 terms tested integers up to 3104, about four times the maximum value of the 57 terms. There is no proof that this condition is sufficient.
%C The terms in the b-file were confirmed up to 500000; for each value of n in 1..710, there is no larger number, up to 500000, than the one listed in the b-file that can be written in the defined form in exactly n ways.
%H Lei Zhou, <a href="/A238266/b238266.txt">Table of n, a(n) for n = 1..710</a>
%e A238263(2)=A238263(3)=1, Max[2,3]=3, so a(1)=3.
%e ...
%e A238263(50)=A238263(51)=...=A238263[71]=18, Max[50, 51,...,71]=71, so a(18)=71.
%t n = 1; sh = {}; target = 57; Do[AppendTo[sh, 0], {i, 1, target}]; While[n < (4*Max[sh] + 100), n++; ct = 0; Do[If[f1 = FactorInteger[i]; l1 = Length[f1]; If[f1[[1, 1]] == 2, l1--]; f2 = FactorInteger[n - i]; l2 = Length[f2]; If[f2[[1, 1]] == 2, l2--]; (l1 <= 1) && (l2 <= 1), ct++], {i, 1, Floor[n/2]}]; If[ct <= target, sh[[ct]] = n;]]; sh
%Y Cf. A000961, A238263, A238264.
%K nonn
%O 1,1
%A _Lei Zhou_, Feb 21 2014
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