A366330 Minimal numbers (with respect to division) with no coprime divisor shift.

2, 15, 33, 51, 69, 87, 123, 141, 159, 177, 213, 249, 267, 303, 321, 339, 393, 411, 447, 501, 519, 537, 573, 591, 665, 681, 699, 717, 753, 771, 789, 807, 819, 843, 879, 933, 951, 1015, 1041, 1059, 1077, 1149, 1167, 1203, 1235, 1257, 1293, 1329, 1347, 1383

Offset

1,1

Comments

A number k has a coprime divisor shift s if GCD(d + s, k) = 1 for all divisors d of k.

A number k has a coprime divisor shift iff it is not divisible by any number in the sequence.

If k has no coprime divisor shift, then so is any multiple of k.

Theorem: if p is an odd prime of the form p = 3*q - 1, then 3*p is a term. Proof: p <> 3 and 3*p has 4 divisors: 1, 3, p, 3*p. A proper divisor of 3*p is either 1 or an odd prime and thus has a coprime divisor shift and is not a term. There are 3 distinct residues of the divisors of 3*p modulo 3, i.e. 0, 1, 2, which is a complete set of residues and thus 3*p has no coprime divisor shift (see A366251). The converse is not true, for instance 819 = 3^2*7*13 is a term. - Chai Wah Wu, Oct 19 2025

References

a(1) = 2 for GCD(2 + 0, 2) > 1 and GCD(1 + 1, 2) > 1.

a(2) = 15 for GCD(3 + 0, 15) > 1, GCD(5 + 1, 15) > 1, GCD(1 + 2, 15) > 1, and any odd number between 2 and 15 has a coprime divisor shift.

Links

M. Farrokhi D. G., Table of n, a(n) for n = 1..10000

Prog

(Python)
from itertools import count, islice
from sympy import divisors, primerange
def A366330_gen(): # generator of terms
    yield 2
    a = []
    for k in count(3, 2):
        ds = divisors(k)
        if all(k%b for b in a) and any(len({d%p for d in ds})==p for p in primerange(3, len(ds)+1)):
            yield k
            a.append(k)
A366330_list = list(islice(A366330_gen(), 50)) # Chai Wah Wu, Oct 19 2025

Crossrefs

Cf. A003627, A044135, A044516, A366219, A366251.

Sequence in context: A196263 A045486 A243469 * A300431 A300765 A300635

Adjacent sequences: A366327 A366328 A366329 * A366331 A366332 A366333

Keyword

nonn

Author

M. Farrokhi D. G., Oct 07 2023

Status

approved