1,4

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.

Ken Clements, Table of n, a(n) for n = 1..82

a(1) = 0 because A027856(1) = 4 = 2^2 * 3^0

a(2) = 1 because A027856(2) = 6 = 2^1 * 3^1

a(3) = 1 because A027856(3) = 12 = 2^2 * 3^1

a(4) = 2 because A027856(4) = 18 = 2^1 * 3^2

a(5) = 2 because A027856(5) = 72 = 2^3 * 3^2

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 *)

(Python)

from math import log10

from gmpy2 import is_prime

l2, l3 = log10(2), log10(3)

upto_digits = 200

sum_limit = 2 + int((upto_digits - l3)/l2)

def TP_pi_2_upto_sum(limit): # Search all partitions up to the given exponent sum.

unsorted_result = [(0, log10(4)), (1, log10(6))]

for exponent_sum in range(3, limit+1, 2):

for i in range(1, exponent_sum):

j = exponent_sum - i

log_N = i*l2 + j*l3

if log_N <= upto_digits:

N = 2**i * 3**j

if is_prime(N-1) and is_prime(N+1):

unsorted_result.append((j, log_N))

sorted_result = sorted(unsorted_result, key=lambda x: x[1])

return sorted_result

print([j for j, _ in TP_pi_2_upto_sum(sum_limit) ])

(Python)

from itertools import islice

from heapq import heappop, heappush

from sympy import isprime, multiplicity

def A386730_gen(): # generator of terms

yield from (0, 1)

h, hset = [2, 3], {2, 3}

while True:

m = heappop(h)

if isprime(m-1) and isprime(m+1):

yield multiplicity(3, m)

for p in (4, 6, 9):

mp = m*p

if mp not in hset:

heappush(h, mp)

hset.add(mp)

A386730_list = list(islice(A386730_gen(), 50)) # Chai Wah Wu, Apr 04 2026

Cf. A027856, A385433, A386731.

Sequence in context: A241568 A335017 A047972 * A004595 A071681 A135621

Adjacent sequences: A386727 A386728 A386729 * A386731 A386732 A386733

nonn,changed

Ken Clements, Jul 31 2025

approved