The base 2 analog of A086766

[sweety439]

In base 2 (in fact, in any base), for the n which have the form (1(0^i))^j (where x^y means a string of y x's in this base) with i>=1, j>=1 (such sequence is 10, 100, 1000, 1010, 10000, 100000, 100100, 101010, 1000000, 100000000, 10001000, 10101010, 1000000000, 100100100, 1000000000, 1000010000, 10000000000, 100000000000, 100000100000, 1000010000, 100010001000, 100100100100, 101010101010, ... sequence is not currently in OEIS, and the base 2 analog sequence is also not in OEIS, but the base 2 analog plus the Mersenne numbers (i.e. of the form 2^n-1) is the OEIS sequence A272919, i.e. delete the Mersenne numbers form the sequence A272919, it become the base 2 analog of this sequence, and convert these numbers from to base 2 to base 10, we get this sequence, also the even numbers in A272919 are exactly the base 2 analog of this sequence, the sequence in base 10 plus the repunit numbers is also not in OEIS), there is at most one prime for this n (the prime is in A252491 or A128889 for the corresponding base, i.e. Phi(p^r,b) for base b, where Phi is the cyclotomic polynomial, and p is prime, r>=2, for base 10, the only known such prime is 101 = Phi(4,10), thus in base 10, it is conjectured that all A086766(n) are 0 for all such n except 10, and it is known that A086766(n) = 0 for all such n <= 10^400 (although the comment in A086766 only says n <= 10^275) (the comment "Conjecture: If n is not of the form 10^m then a(n) is nonzero." in A086766 is not true, A086766(1010) = A086766(100100) = A086766(101010) = 0, this conjecture should be "If n is not of the form (1(0^i))^j (where x^y means a string of y x's) then a(n) is nonzero"), note that this conjecture has counterexamples in other bases, in dozenal (base 12), n = 33 in dozenal (39 in decimal) cannot have any primes, since the formula is (36*144^n-25)/11, which can be factored as (6*12^n-5) * (6*12^n+5)/11, thus I doubt that there is also a such counterexample in base 10, and for base 2, the only known such primes are Phi(n,2) for n = 4, 8, 9, 16, 27, 32, 49, 3481 (see https://oeis.org/A297625), and it is conjectured that there are no other such primes, and this is first terms for the base 2 analog of A086766:

1,1
2,1
3,1
4,2
5,1
6,1
7,2
8,1
9,1
10,impossible (10 = 1010 in base 2)
11,1
12,2
13,2
14,1
15,1
16,impossible (16 = 10000 in base 2, the only possibility is Phi(25,2) = 1082401, but it is not prime)
17,2
18,1
19,4
20,1
21,1
22,2
23,1
24,9
25,18
26,1
27,2
28,6
29,1
30,1
31,6
32,impossible (32 = 100000 in base 2)
33,1
34,2
35,1
36,1
37,1136
38,4
39,1
40,3
41,1
42,impossible (42 = 101010 in base 2)
43,2
44,1
45,2
46,2
47,126
48,1
49,5
50,1
51,1
52,2
53,1
54,1
55,2
56,1
57,2
58,2
59,6
60,3
61,18
62,6
63,1
64,6
65,1
66,2
67,4
68,1
69,1
70,2
71,6
72,4
73,unknown (n=73 has been searched to 5000 with no prime or PRP found, can someone find it?)
74,1
75,1
76,2
77,2
78,1
79,4
80,2
81,1
82,2
83,1
84,2
85,48
86,1
87,2
88,4
89,1
90,1
91,4
92,46
93,29
94,24
95,1
96,1
97,unknown (n=97 has been searched to 5000 with no prime or PRP found, can someone find it?)
98,1
99,1
100,2
101,3
102,2
103,20
104,2
105,1
106,10
107,402
108,3
109,2
110,6
111,1
112,56
113,1
114,1
115,2
116,1
117,2
118,48
119,1
120,1
121,2
122,2
123,4
124,16
125,1
126,36
127,18
128,1
129,5
130,2
131,1
132,4
133,5
134,1
135,1
136,impossible (136 = 10001000 in base 2)