A385433 -id:A385433 - cp's OEIS frontend

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

Ken Clements, Jul 31 2025

nonn

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.

			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

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

Cf. A385433, A386730, A027856.

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 *)

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) ])

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

Original entry on oeis.org

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

Ken Clements, Jul 31 2025

nonn

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.

			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

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

Cf. A027856, A385433, A386731.

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 *)

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) ])

A386857 Numbers k such that both 92^k - 1 and 92^k + 1 are prime.

Original entry on oeis.org

1, 3, 7, 43, 63, 211

Offset: 1

Ken Clements, Aug 05 2025

The exponent, k, of 2 must be odd because the exponent, 2, of 3 (where 9 = 3^2) is even and the sum of the exponents of prime factors 2 and 3 must be odd to form a product that is a twin prime average. Of all subsequences of A027856, this is the longest known where the power of 3 is fixed.

Amiram Eldar noted that using A002236 and A002256, we obtain a(7) > 5.6*10^6, if it exists.

			a(1) = 1 because 2*9 = 18 with 17 and 19 prime.
a(2) = 3 because 8*9 = 72 with 71 and 73 prime.
a(3) = 7 because 128*9 = 1152 with 1151 and 1153 prime.
a(4) = 43 because 8796093022208*9 = 79164837199872 with 79164837199871 and 79164837199873 prime.

Intersection of A002236 and A002256.

Cf. A014574, A384530, A027856, A385433, A181490.

q:= k-> (m-> andmap(isprime, [m-1, m+1]))(9*2^k):
select(q, [2*i-1$i=1..111])[];  # Alois P. Heinz, Aug 08 2025

from gmpy2 import is_prime
def is_TPpi2(e2, e3):
    N = 2**e2 * 3**e3
    return is_prime(N-1) and is_prime(N+1)
print([k for k in range(1, 100001, 2) if is_TPpi2(k, 2)])

cp's OEIS Frontend

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

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

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A386857 Numbers k such that both 92^k - 1 and 92^k + 1 are prime.

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cp's OEIS Frontend

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

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Examples

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Mathematica

Python

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A386857 Numbers k such that both 9*2^k - 1 and 9*2^k + 1 are prime.

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A386857 Numbers k such that both 92^k - 1 and 92^k + 1 are prime.