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URL: https://oeis.org/A384835

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A384835
The exponents (j, k) of the numbers 2^j*3^k that are averages of twin primes, with both j and k > 0, in the order of their sum, and then by j.
1
1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 4, 3, 6, 1, 5, 4, 7, 2, 3, 10, 6, 7, 2, 15, 12, 5, 18, 1, 18, 5, 21, 4, 24, 5, 27, 4, 11, 24, 30, 7, 32, 9, 33, 8, 31, 12, 36, 7, 43, 2, 32, 15, 43, 8, 50, 9, 63, 2, 66, 25, 79, 20, 99, 10, 57, 64, 82, 63, 63, 88, 56, 99, 148, 27
OFFSET
1,4
COMMENTS
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.
LINKS
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A027856, A384639 (ordered by value of 2^j*3^k).
Sequence in context: A307319 A131730 A029335 * A029257 A194258 A165927
KEYWORD
nonn,tabf
AUTHOR
Ken Clements, Jun 10 2025
STATUS
approved