A344385 Numbers k of least prime signature (A025487) such that k-1 or k+1 is prime.

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 72, 96, 128, 180, 192, 210, 240, 256, 360, 384, 420, 432, 480, 576, 720, 768, 840, 864, 1152, 1260, 1296, 1440, 1800, 2160, 2310, 2520, 2592, 2880, 3360, 3456, 4620, 5040, 5400, 6144, 6300, 6480, 6720, 6912, 7560

Offset

1,2

Comments

The corresponding prime numbers are A344384.

64 is the first number of least prime signature not in this sequence.

Questions: 1) Is this sequence infinite? 2) Is log(a(n)) = O(log(n)^2)? For the first 132 terms, log(a(n)) is fit unusually well by a quadratic polynomial in log(n).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..132 from Hal M. Switkay)

Example

16 is a term because 16 is in A025487 and 16 + 1 is prime.

Mathematica

(* Load the function f[n] at A025487, then: *)
Select[Union@ Flatten@ f[6], AnyTrue[# + {-1, 1}, PrimeQ] &]
(* or use the b-file at A025487: *)
Select[TakeWhile[Import["https://oeis.org/A025487/b025487.txt", "Data"], Length@ # == 2 &][[All, -1]], AnyTrue[# + {-1, 1}, PrimeQ] &] (* Michael De Vlieger, May 16 2021 *)

Prog

(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import isprime, factorint, prevprime, nextprime
def A344385_gen(): # generator of terms
    h, hset = [1], {1}
    while True:
        m = heappop(h)
        if isprime(m-1) or isprime(m+1):
            yield m
        ps = factorint(m)
        for p in ps:
            if p == 2 or ps[prevprime(p)]>ps[p]:
                mp = m*p
                if mp not in hset:
                    heappush(h, mp)
                    hset.add(mp)
        mp = m*nextprime(max(ps.keys(), default=1))
        if mp not in hset:
            heappush(h, mp)
            hset.add(mp)
A344385_list = list(islice(A344385_gen(), 40)) # Chai Wah Wu, Mar 23 2026

Crossrefs

Cf. A000040, A025487, A344384.

Sequence in context: A385722 A279537 A325238 * A070175 A096850 A250270

Adjacent sequences: A344382 A344383 A344384 * A344386 A344387 A344388

Keyword

nonn

Author

Hal M. Switkay, May 16 2021

Status

approved