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URL: https://oeis.org/A386730

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A386730
a(n) is the 3-adic valuation of A027856(n).
3
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
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
KEYWORD
nonn,changed
AUTHOR
Ken Clements, Jul 31 2025
STATUS
approved