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A387095 a(1) = 1 for the single prime 3; for n>=2, a(n) is the number of primes between 2^n and 2^(n+1) whose pairs lay symmetrically at each side of the center 3*2^(n-1) of that interval.

%I A387095 #42 Aug 21 2025 16:12:39
%S A387095 1,2,2,4,4,6,8,22,26,42,92,128,218,416,750,1300,2342,4136,7440,13572,
%T A387095 24820,45420,82922,152964,282626,522354,972388,1809744,3379508,
%U A387095 6318652,11855790,22277960,41917480
%N A387095 a(1) = 1 for the single prime 3; for n>=2, a(n) is the number of primes between 2^n and 2^(n+1) whose pairs lay symmetrically at each side of the center 3*2^(n-1) of that interval.
%C A387095 These primes form the sequence A092570.
%C A387095 With a different approach by seeking symmetries in the distribution of primes, we arrive at the same sequence as A092570. Here, the numbers of primes that fit those symmetries in their ranges form this sequence. Those ranges, interestingly, lay between the powers of 2.
%C A387095 The sum of each prime pair in a range is 3*2^n. (Sequence A007283).
%C A387095 In rows 2, 3, and 7, the middle primes (5, 7), (11, 13) and (191, 193) respectively are just next to the centers of their ranges, and with gaps of size 2 in between them, they are twin prime pairs.
%C A387095 If it could be proved that a derived sequence, 0, 1, 1, 5, 5, 7, 1, ..., that is the distances of the closest primes to the centers 3*2^n of the ranges contains infinitely many 1's, which always correspond to twin primes, then that would also prove the infinitude of twin primes. Of that particular form, A181494 shows the only known eight primes.
%e A387095 Intervals:                         Primes                       Centers       Count
%e A387095                                 in A092570:                  of intervals:    a(n):
%e A387095   [2,4]                              3                             3            1
%e A387095   [4,8]                           5     7                          6            2
%e A387095   [8,16]                         11    13                         12            2
%e A387095   [16,32]                  17    19    29    31                   24            4
%e A387095   [32,64]                  37    43    53    59                   48            4
%e A387095   [64,128]           79    83    89   103   109   113             96            6
%e A387095   [128,256]   151   157   173   191   193   211   227   233      192            8
%e A387095   ...
%e A387095 In row 5 for the interval 2^5 = 32 to 2^6 = 64, the prime pair 37 and 59 lay symmetrically to each side of the center of that interval, 3*2^4 = 48 as abs(48-37) = abs(48-59) = 11, and their sum 37 + 59 = 96, which is 3*2^5. So are 43 and 53 in similarly symmetrical positions in that range, with their sum being 96 also.
%t A387095 a[n_] := Module[{c = 0, r = r1 = 3*2^(n-1)}, While[(r1 = NextPrime[r1, -1]) > 2^n, If[PrimeQ[2*r - r1], c += 2]]; c]; a[1] = 1; Array[a, 20] (* _Amiram Eldar_, Aug 16 2025 *)
%Y A387095 Cf. A000040, A007283, A092570, A181494.
%K A387095 nonn,more,new
%O A387095 1,2
%A A387095 _Tamas Sandor Nagy_, Aug 16 2025
%E A387095 a(13)-a(33) from _Amiram Eldar_, Aug 16 2025