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

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, 20, 10, 64, 63, 27, 88, 99, 2, 16, 10, 169, 135, 51, 141, 52, 231, 320, 44, 419, 143, 476, 207, 332, 97, 324, 738, 1493, 1320, 333, 1167, 188, 1440, 2251, 2033

Offset

1,4

Comments

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.

Links

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

Example

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A027856, A385433, A386731.

Sequence in context: A241568 A335017 A047972 * A004595 A071681 A135621

Adjacent sequences: A386727 A386728 A386729 * A386731 A386732 A386733

Keyword

nonn

Author

Ken Clements, Jul 31 2025

Status

approved