A386252 Numbers m of the form 2^i * 3^j * 5^k such that i, j, k > 0 and m+1 and m-1 are both prime numbers.

30, 60, 150, 180, 240, 270, 600, 810, 1620, 3000, 4050, 4800, 9000, 9720, 15360, 21600, 23040, 33750, 138240, 180000, 281250, 345600, 737280, 3456000, 6144000, 6561000, 10125000, 13668750, 15552000, 17496000, 20995200, 22118400, 24000000, 30000000, 54675000

Offset: 1

Author

Ken Clements, Jul 16 2025

Keywords

• nonn

Examples

			a(1) = 2^1 * 3^1 * 5^1 = 30 where 29 and 31 are prime numbers.
a(2) = 2^2 * 3^1 * 5^1 = 60 where 59 and 61 are prime numbers.
a(3) = 2^1 * 3^1 * 5^2 = 150 where 149 and 151 are prime numbers.
a(4) = 2^2 * 3^2 * 5^1 = 180 where 179 and 181 are prime numbers.
		

Links

• Ken Clements, Table of n, a(n) for n = 1..926

Crossrefs

Subsequence of A143207.

Cf. A027856, A384530, A080185.

Programs

Mathematica

seq[max_] := Select[Table[2^i*3^j*5^k, {i, 1, Log2[max]}, {j, 1, Log[3, max/2^i]}, {k, 1, Log[5, max/(2^i*3^j)]}] // Flatten // Sort, And @@ PrimeQ[# + {-1, 1}] &]; seq[10^8] (* Amiram Eldar, Jul 17 2025 *)

Python

from math import log10
from gmpy2 import is_prime
l2, l3, l5 = log10(2), log10(3), log10(5)
upto_digits = 20
sum_limit = 3 + int((upto_digits - l3 - l5)/l2)
def TP_pi_3_upto_sum(limit): # Search all partitions up to the given exponent sum.
    unsorted_result = []
    for exponent_sum in range(3, limit+1):
        for i in range(1, exponent_sum -1):
             for j in range(1, exponent_sum - i):
                k = exponent_sum - i - j
                log_N = i*l2 + j*l3 + k*l5
                if log_N <= upto_digits:
                    N = 2**i * 3**j * 5**k
                    if is_prime(N-1) and is_prime(N+1):
                        unsorted_result.append((N, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([n for n, _ in TP_pi_3_upto_sum(sum_limit) ])