A384337 Numbers k such that there exists m > k with k | m^3 + 1 and m | k^3 + 1.

1, 2, 3, 5, 9, 14, 35, 45, 49, 54, 61, 65, 93, 99, 114, 117, 146, 147, 185, 234, 299, 325, 329, 362, 365, 398, 413, 434, 437, 549, 594, 619, 626, 635, 794, 874, 915, 962, 981, 1057, 1209, 1251, 1550, 1638, 1699, 2021, 2110, 2149, 2219, 2345, 2394, 2409, 2449, 2667, 2763, 2771, 2881, 2989, 3002

Offset

1,2

Comments

If k is a term with m > k, k | m^3 + 1 and m | k^3 + 1, then m is also a term, as with s = (m^3 + 1)/k we have s > m (in fact s > m^2), s | m^3 + 1 and m | s^3 + 1. In particular, the sequence is infinite.

Are there any terms divisible by 4?

Links

Robert Israel, Table of n, a(n) for n = 1..950

Mathematics StackExchange, The cubed plus 1 problem

Example

a(4) = 5 is a term because both m = 9 and m = 14 work: 5 | 9^3 + 1 = 730 and 9 | 5^3 + 1 = 126, 5 | 14^3 + 1 = 2745 and 14 | 5^3 + 1 = 126.

Maple

filter:= proc(n) ormap(t -> t > n and t^3 + 1 mod n = 0, numtheory:-divisors(n^3+1)) end proc:
select(filter, [$1 .. 10000]);

Prog

(Python)
from itertools import count, islice
from sympy import divisors
def A384337_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue, 1)):
        for m in divisors(k**3+1, generator=True):
            if m>k and not (m**3+1)%k:
                yield k
                break
A384337_list = list(islice(A384337_gen(), 30)) # Chai Wah Wu, Jun 02 2025

Crossrefs

Sequence in context: A105044 A026008 A101461 * A085897 A361906 A306829

Adjacent sequences: A384334 A384335 A384336 * A384338 A384339 A384340

Keyword

nonn

Author

Robert Israel, May 26 2025

Status

approved