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%I #23 Jun 12 2014 18:50:47
%S 2,2,2,2,2,2,4,2,4,2,2,3,6,6,4,2,3,8,6,5,9,5,9,2,6,10,9,9,8,6,13,8,7,
%T 13,8,10,15,5,13,12,17,13,8,9,13,13,15,17,13,10,20,10,14,19,14,21,14,
%U 13,14,14,14,20,10,20,16,25,15,18,21,16,20,22,13,17
%N The number of integers that can be written in the form 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 a(n) > 0 for all n > 0.
%C The Mathematica program used to generate the first 74 terms tested integers up to 5048, about four times the maximum number that was found to have fewer than 75 ways as defined.
%C The establishment of this sequence depends on A238266, which limits the search range of this sequence.
%C The b-file is calculated by evaluating integers from 1 to 500000, more than 12 times the maximum number that can be written in the defined form in no more than 710 ways, as of A238266.
%H Lei Zhou, <a href="/A238267/b238267.txt">Table of n, a(n) for n = 1..710</a>
%e A238263(2)=A238263(3)=1; these two numbers, 2 and 3, are the only numbers that can be written in the defined form in only one way, so a(1)=2.
%e ...
%e A238263(50)=A238263(51)=...=A238263(71)=18; 8 numbers, {50, 51, 55, 58, 59, 61, 67, 71}, were found to have 18 ways to be written in the defined form, so a(18)=8.
%t n = 1; sc = {}; max = 0; target = 74; Do[AppendTo[sc, 0], {i, 1, target}]; While[n < (4*max + 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, sc[[ct]]++; max = n]]; sc
%Y Cf. A000961, A238263, A238264, A238266.
%K nonn
%O 1,1
%A _Lei Zhou_, Feb 21 2014
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