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A386730 a(n) is the 3-adic valuation of A027856(n).

%I A386730 #12 Aug 04 2025 01:11:43
%S A386730 0,1,1,2,2,3,1,3,2,4,7,10,1,5,15,5,4,5,4,7,8,2,9,7,24,12,8,15,9,2,25,
%T A386730 20,10,64,63,27,88,99,2,16,10,169,135,51,141,52,231,320,44,419,143,
%U A386730 476,207,332,97,324,738,1493,1320,333,1167,188,1440,2251,2033
%N A386730 a(n) is the 3-adic valuation of A027856(n).
%C A386730 These are the exponents, j, of the prime factor 3 of the A027856 numbers m = 2^i * 3^j where m is the average of twin primes. Except for the first term, all are greater than zero because all other A027856 numbers have both 2 and 3 as prime factors. After the second term, all sums of i+j are odd because even sums make either m-1 or m+1 divisible by 5, which precludes twin primes except for the case of 6, where m-1 is divisible by 5, but 5 is the only number divisible by 5 that is also prime.
%H A386730 Ken Clements, <a href="/A386730/b386730.txt">Table of n, a(n) for n = 1..82</a>
%e A386730 a(1) = 0 because A027856(1) = 4 = 2^2 * 3^0
%e A386730 a(2) = 1 because A027856(2) = 6 = 2^1 * 3^1
%e A386730 a(3) = 1 because A027856(3) = 12 = 2^2 * 3^1
%e A386730 a(4) = 2 because A027856(4) = 18 = 2^1 * 3^2
%e A386730 a(5) = 2 because A027856(5) = 72 = 2^3 * 3^2
%t A386730 seq[max_] := IntegerExponent[Select[Sort[Flatten[Table[2^i*3^j, {i, 1, Floor[Log2[max]]}, {j, 0, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {-1, 1}] &], 3]; seq[10^250] (* _Amiram Eldar_, Aug 01 2025 *)
%o A386730 (Python)
%o A386730 from math import log10
%o A386730 from gmpy2 import is_prime
%o A386730 l2, l3 = log10(2), log10(3)
%o A386730 upto_digits = 200
%o A386730 sum_limit = 2 + int((upto_digits - l3)/l2)
%o A386730 def TP_pi_2_upto_sum(limit): # Search all partitions up to the given exponent sum.
%o A386730     unsorted_result = [(0, log10(4)), (1, log10(6))]
%o A386730     for exponent_sum in range(3, limit+1, 2):
%o A386730         for i in range(1, exponent_sum):
%o A386730             j = exponent_sum - i
%o A386730             log_N = i*l2 + j*l3
%o A386730             if log_N <= upto_digits:
%o A386730                 N = 2**i * 3**j
%o A386730                 if is_prime(N-1) and is_prime(N+1):
%o A386730                      unsorted_result.append((j, log_N))
%o A386730     sorted_result = sorted(unsorted_result, key=lambda x: x[1])
%o A386730     return sorted_result
%o A386730 print([j for j, _ in TP_pi_2_upto_sum(sum_limit) ])
%Y A386730 Cf. A027856, A385433, A386731.
%K A386730 nonn
%O A386730 1,4
%A A386730 _Ken Clements_, Jul 31 2025