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

1,1

Links

Chai Wah Wu, Table of n, a(n) for n = 1..6085 (terms 1..926 from Ken Clements)

Example

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.

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 *)

Prog

(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) ])
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import isprime
def A386252_gen(): # generator of terms
    h, hset = [30], {30}
    while True:
        m = heappop(h)
        if isprime(m-1) and isprime(m+1):
            yield m
        for p in (2, 3, 5):
            mp = m*p
            if mp not in hset:
                heappush(h, mp)
                hset.add(mp)
A386252_list = list(islice(A386252_gen(), 45)) # Chai Wah Wu, Apr 21 2026

Crossrefs

Subsequence of A143207.

Cf. A027856, A384530, A080185.

Sequence in context: A359410 A108454 A235483 * A387673 A064783 A228877

Adjacent sequences: A386249 A386250 A386251 * A386253 A386254 A386255

Keyword

nonn

Author

Ken Clements, Jul 16 2025

Status

approved