A386731 -id:A386731 - cp's OEIS frontend

A385433 a(n) is the 2-adic valuation of A027856(n).

2, 1, 2, 1, 3, 2, 6, 4, 7, 5, 6, 3, 18, 12, 2, 18, 21, 24, 27, 30, 33, 43, 32, 36, 11, 31, 43, 32, 50, 63, 66, 79, 99, 57, 82, 148, 63, 56, 211, 275, 287, 90, 148, 298, 160, 363, 134, 49, 529, 264, 960, 541, 988, 1015, 1440, 1295, 979, 258, 637, 2320, 1036, 2815, 1063, 180, 888

Offset: 1

Ken Clements, Jul 31 2025

nonn

These are the exponents, i, of the prime factor 2 of the A027856 numbers m = 2^i * 3^j where m is the average of twin primes. After the second term, all sums of i+j are odd because even sums (where j>0) 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) = 2 because A027856(1) = 4 = 2^2 * 3^0
a(2) = 1 because A027856(2) = 6 = 2^1 * 3^1
a(3) = 2 because A027856(3) = 12 = 2^2 * 3^1
a(4) = 1 because A027856(4) = 18 = 2^1 * 3^2
a(5) = 3 because A027856(5) = 72 = 2^3 * 3^2

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

Cf. A027856, A386730, 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}] &], 2]; 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, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([i for i, _ 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) ])

cp's OEIS Frontend

A385433 a(n) is the 2-adic valuation of A027856(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python