A386731 a(n) = A385433(n) + A386730(n).

2, 2, 3, 3, 5, 5, 7, 7, 9, 9, 13, 13, 19, 17, 17, 23, 25, 29, 31, 37, 41, 45, 41, 43, 35, 43, 51, 47, 59, 65, 91, 99, 109, 121, 145, 175, 151, 155, 213, 291, 297, 259, 283, 349, 301, 415, 365, 369, 573, 683, 1103, 1017, 1195, 1347, 1537, 1619, 1717, 1751, 1957

Offset

1,1

Comments

These numbers are sum of the exponents of 2 and 3 for the averages of twin primes in A027856. An interesting aspect is that after the first 2 terms, all of these are odd numbers. For all of those, the sum cannot be even because then for m = 2^i * 3^j, m-1 or m+1 would be divisible by 5.

Links

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

Example

a(1) = A385433(1) + A386730(1) = 2
a(2) = A385433(2) + A386730(2) = 2
a(3) = A385433(3) + A386730(3) = 3
a(4) = A385433(4) + A386730(4) = 3
a(5) = A385433(5) + A386730(5) = 5

Mathematica

seq[max_] := Total[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}] &], #] & /@ {2, 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 = [(2, 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((i+j, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([s for s, _ in TP_pi_2_upto_sum(sum_limit) ])
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import isprime, multiplicity
def A386731_gen(): # generator of terms
    yield from (2, 2)
    h, hset = [2, 3], {2, 3}
    while True:
        m = heappop(h)
        if isprime(m-1) and isprime(m+1):
            yield (~m&m-1).bit_length()+multiplicity(3, m)
        for p in (4, 6, 9):
            mp = m*p
            if mp not in hset:
                heappush(h, mp)
                hset.add(mp)
A386731_list = list(islice(A386731_gen(), 50)) # Chai Wah Wu, Apr 04 2026

Crossrefs

Cf. A385433, A386730, A027856.

Sequence in context: A283529 A064986 A349675 * A029019 A040039 A008667

Adjacent sequences: A386728 A386729 A386730 * A386732 A386733 A386734

Keyword

nonn

Author

Ken Clements, Jul 31 2025

Status

approved