A385935 Numbers that are one less than terms of A055932, and also prime.

3, 5, 7, 11, 17, 23, 29, 31, 47, 53, 59, 71, 89, 107, 127, 149, 179, 191, 239, 269, 359, 383, 419, 431, 449, 479, 599, 647, 719, 809, 839, 863, 971, 1049, 1151, 1259, 1439, 1499, 1619, 1889, 2099, 2309, 2399, 2591, 2699, 2879, 2939, 2999, 3359, 3779, 4049, 4373, 4409, 4799

Offset

1,1

Links

Table of n, a(n) for n=1..54.

Example

a(1) = A055932(3)-1 = 4-1 = 3
a(2) = A055932(4)-1 = 6-1 = 5
a(3) = A055932(5)-1 = 8-1 = 7
a(4) = A055932(6)-1 = 12-1 = 11
a(5) = A055932(8)-1 = 18-1 = 17
a(6) = A055932(9)-1 = 24-1 = 23
a(7) = A055932(10)-1 = 30-1 = 29
a(8) = A055932(11)-1 = 32-1 = 31

Mathematica

Select[Prime[Range[1000]], (p = FactorInteger[#+1][[;; , 1]])[[-1]] == Prime[Length[p]] &] (* Amiram Eldar, Jul 13 2025 *)

Prog

(Python)
from sympy import primefactors, prime, isprime
def is_pi_complete(n):  # n is a term of A055932
    factors = primefactors(n)
    return factors[-1] == prime(len(factors))
def aupto(limit):
    return [n-1 for n in range(4, limit+1, 2) if isprime(n-1) and is_pi_complete(n)]
print(aupto(100_000))
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import factorint, isprime, primefactors, nextprime
def A385935_gen(): # generator of terms
    def is_squarefree(n): return max(factorint(n).values(), default=1)<=1
    h, hset = [1], {1}
    while True:
        m = heappop(h)
        if isprime(m-1):
            yield m-1
        ps = primefactors(m)
        ps.append(nextprime(max(ps, default=1)))
        for p in ps:
            mp = m*p
            if mp not in hset:
                heappush(h, mp)
                hset.add(mp)
A385935_list = list(islice(A385935_gen(), 40)) # Chai Wah Wu, Mar 21 2026

Crossrefs

Cf. A055932, A385934.

Sequence in context: A331891 A331896 A108539 * A170886 A361823 A238137

Adjacent sequences: A385932 A385933 A385934 * A385936 A385937 A385938

Keyword

nonn

Author

Ken Clements, Jul 12 2025

Status

approved