A389863 Numbers k with exactly two distinct repeated values among the fractions (2*k mod m)/m for 1 <= m <= k.

14, 15, 16, 21, 22, 26, 27, 30, 34, 36, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482

Offset

1,1

Comments

Conjecture: All numbers of the form 2*p with p prime and p > 5 belong to the sequence, and no other numbers greater than 36 do. Empirically verified for k <= 100000.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Example

For k = 12, the fractions for 1 <= m <= 12 are 0, 0, 0, 0, 4/5, 0, 3/7, 0, 2/3, 2/5, 2/11, 0. Only 0 is repeated (one alignment), so 12 is not in this sequence.
For k = 14, the fractions 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. Exactly two values are repeated (0 and 1/3), so 14 is in this sequence.

Maple

filter:= proc(k) local m, t, S, R, s;
  t:= 0; S:= {}: R:= {}:
  for m from 1 to k do
    s:= (2*k mod m)/m;
    if member(s, S) then
      if not member(s, R) then
        t:= t+1; if t = 3 then return false fi;
        R:= R union {s}
      fi
    else S:= S union {s}
    fi
  od;
  t = 2
end proc:
select(filter, [$1..600]); # Robert Israel, Oct 19 2025

Mathematica

okQ[k_]:=Module[{fr={}}, Do[AppendTo[fr, Mod[2k, m]/m], {m, 1, k}]; Length[Select[Last/@Tally[fr], #>1&]]==2]; Select[Range[482], okQ] (* James C. McMahon, Nov 01 2025 *)

Prog

(Python)
from fractions import Fraction
def ok(k, target=2):
    seen, dup = set(), set()
    for m in range(1, k + 1):
        f = Fraction((2 * k) % m, m)
        if f in seen:
            dup.add(f)
        else:
            seen.add(f)
    return len(dup) == target
target = 2
for k in range(2, 300):
    if ok(k, target):
        print(k)
(Python)
from gmpy2 import mpq
from collections import Counter
def ok(n):
    C, F = set(), set()
    for m in range(1, n+1):
        f = mpq(2*n%m, m)
        if f in F:
            C.add(f)
            if len(C) > 2: return False
        else:
            F.add(f)
    return len(C) == 2
print([k for k in range(1, 483) if ok(k)]) # Michael S. Branicky, Oct 23 2025

Crossrefs

Cf. A387426, A100484.

Sequence in context: A163482 A257782 A206447 * A293792 A308974 A130687

Adjacent sequences: A389860 A389861 A389862 * A389864 A389865 A389866

Keyword

nonn

Author

Kenneth J Scheller, Oct 17 2025

Status

approved