A387319 Numbers k for which modulo 3 and modulo another multiple of 3 are coincident among the fractions (2*k mod m)/m for 1 <= m <= k.

6, 9, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset

1,1

Comments

Conjecture: Every composite number, except 4, 8, 10, and 25, has at least one coincident pair linking m = 3 with another multiple of 3 among the fractions (2*k mod m)/m for 1 <= m <= k, while no prime has such a pair. Empirically verified to 10^5.

Additional observation: For even k, the first (3, multiple-of-3) coincidence occurs at m = 12, while for odd k == 1 or 5 (mod 6), the corresponding coincidences vary and appear related to the factorization pattern of k.

Conjecture (based on comparing the first 10,000 terms): a(n) = A343311(n) for n>=3. - R. J. Mathar, Nov 10 2025

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Example

For k = 13, the fractions (2*k mod m)/m for 1 <= m <= 13 are 0, 0, 2/3, 1/2, 1/5, 1/3, 5/7, 1/4, 8/9, 3/5, 4/11, 1/6, 0. The only repeated value is 0 (at m = 1, 2, 13). No repeated value occurs between m = 3 and any larger multiple of 3, since the fractions at m = 3, 6, 9, 12 (2/3, 1/3, 8/9, 1/6) are all distinct. Hence 13 is not in this sequence.
For k = 14, the fractions (2*k mod m)/m for 1 <= m <= 14 are 0, 0, 1/3, 0, 3/5, 2/3, 0, 1/2, 1/9, 4/5, 6/11, 1/3, 2/13, 0. Two values repeat: 0 (at m = 1, 2, 4, 7, 14) and 1/3 (at m = 3 and 12). The coincidence of 1/3 links m = 3 with another multiple of 3 (12), so k = 14 is in this sequence.

Mathematica

A387319Q[k_] := With[{f = Mod[2*k, 3]/3}, AnyTrue[Range[6, k, 3], Mod[2*k, #]/# == f &]];
Select[Range[100], A387319Q] (* Paolo Xausa, Nov 06 2025 *)

Prog

(Python)
from fractions import Fraction
def ok(k):
    f3 = Fraction((2*k)%3, 3)
    for m in range(6, k+1, 3):
        if Fraction((2*k)%m, m)==f3:
            return True
    return False
print([k for k in range(1, 200) if ok(k)])

Crossrefs

Cf. A002808 (composites), A000040 (primes), A387426, A389863.

Sequence in context: A185177 A185179 A072546 * A344805 A295670 A272466

Adjacent sequences: A387316 A387317 A387318 * A387320 A387321 A387322

Keyword

nonn

Author

Kenneth J Scheller, Oct 24 2025

Status

approved