1,4

These are the (j,k) exponents of the numbers 2^j*3^k that are averages of twin primes, ordered by j+k, j. They are remarkable in structure because except for the first pair (2^1*3^1), j+k is always an odd number. I have proof of this, and the reason it is not the case for the first pair is that 6-1=5 is the only number divisible by 5 that is prime.

Ken Clements, Table of n, a(n) for n = 1..162

2^a(1) * 3^a(2) = 6.

2^a(3) * 3^a(4) = 18.

2^a(5) * 3^a(6) = 12.

2^a(7) * 3^a(8) = 108.

2^a(9) * 3^a(10) = 72.

seq[max_] := Flatten@ Transpose[IntegerExponent[Select[Flatten[Table[2^j*3^(m-j), {m, 2, max}, {j, 1, m-1}]], And @@ PrimeQ[# + {-1, 1}] &], #] & /@ {2, 3}]; seq[200] (* Amiram Eldar, Jun 26 2025 *)

(Python)

from sympy import isprime

def is_TP_pi_2(j, k):

N = 2**j * 3**k

return isprime(N-1) and isprime(N+1)

def aupto(limit):

result = [1, 1]

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

for j in range(1, exponent_sum):

k = exponent_sum - j

if is_TP_pi_2(j, k):

result.append(j)

result.append(k)

return result

print(aupto(10_000))

(Python)

import heapq

from gmpy2 import is_prime

from itertools import islice

def agen(): # generator of terms

v, oldv, h = 1, 0, [(2, 1, 1, 6)]

while True:

s, e2, e3, v = heapq.heappop(h)

if v != oldv:

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

yield from (e2, e3)

oldv = v

heapq.heappush(h, (s+1, e2+1, e3, 2*v))

heapq.heappush(h, (s+1, e2, e3+1, 3*v))

print(list(islice(agen(), 70))) # Michael S. Branicky, Jun 26 2025

Cf. A027856, A384639 (ordered by value of 2^j*3^k).

Sequence in context: A307319 A131730 A029335 * A029257 A194258 A165927

Adjacent sequences: A384832 A384833 A384834 * A384836 A384837 A384838

nonn,tabf

Ken Clements, Jun 10 2025

approved