Equals 0.04390896...
Let S'(j) = -(Sum_{m=1..j-1} (-1)^m/A002808(m)^2 + (1/2)*(-1)^j/A002808(j)^2) and let c_k be the smallest odd composite > 2^k; S'(c_k) quickly converges to a limit, as illustrated below:
k c_k S'(c_k)
== ======== ===============================
3 9 0.04417438271604938271604938...
4 21 0.04390073853615520282186948...
5 33 0.04390758368090798391978693...
6 65 0.04390888269964319809070094...
7 129 0.04390902395888932501501797...
8 259 0.04390896620540588616012725...
9 513 0.04390896281303069589885533...
10 1025 0.04390896330786777379414334...
11 2049 0.04390896335161701542401577...
12 4097 0.04390896335102793828470954...
13 8193 0.04390896335127457473079624...
14 16385 0.04390896335131185998890588...
15 32769 0.04390896335130880417881285...
16 65541 0.04390896335130852088789156...
17 131073 0.04390896335130852182995244...
18 262145 0.04390896335130852702777625...
19 524289 0.04390896335130852688659318...
20 1048577 0.04390896335130852691520992...
21 2097153 0.04390896335130852691785136...
22 4194305 0.04390896335130852691786707...
23 8388609 0.04390896335130852691787563...
24 16777217 0.04390896335130852691787421...
25 33554433 0.04390896335130852691787435...
...
Extending this several steps farther, it becomes apparent that the limit is 0.04390896335130852691787434869606... (End)
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