A390913 Largest positive integer k such that exactly 1/n of the integers from 1 to k are divisors of k, or 0 if no solution exists.

2, 12, 24, 36, 60, 72, 84, 96, 108, 180, 132, 240, 156, 252, 360, 288, 204, 0, 228, 480, 504, 396, 276, 720, 600, 468, 0, 672, 348, 0, 372, 448, 792, 612, 1260, 864, 444, 684, 936, 1440, 492, 1680, 516, 1056, 0, 828, 564, 1344, 1176, 1800, 1224, 1248, 636, 2160, 1980, 2016, 1368, 1044, 708, 1920

Offset

1,1

Comments

a(n) = 0 for n in A036763. When a(n) > 0, k <= 4*n^2 since d(k) <= 2*sqrt(k) and d(k)*n = k. Here d(k) = A000005(k) is the number of divisors of k.

All nonzero terms are refactorable numbers (A033950).

A353011 is a subsequence. a(11) = 132 is not in A353011 because the earlier a(10) = A353011(10) = 180 is larger.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Formula

a(n) = max({k in Z+ : d(k)*n = k} U {0}), where d(k) = A000005(k).

Example

a(1) = 2 because 2 is the largest k that satisfies d(k) = k/1.
a(2) = 12 because 12 is the largest k that satisfies d(k) = k/2.
a(18) = 0 because no k satisfies d(k) = k/18.

Mathematica

Table[If[IntegerQ[#], #, 0] &@ SelectFirst[Range[4*n^2, 1, -1], DivisorSigma[0, #]*n == # &], {n, 60}] (* Michael De Vlieger, Nov 23 2025 *)

Prog

(Python)
from sympy import divisor_count
def a(n):
    for k in range(4*n*n, 0, -1):
        if divisor_count(k) * n == k:
            return k
    return 0
(Python)
from sympy import divisor_count
def A390913(n): return next((k*n for k in range(n<<2, 0, -1) if divisor_count(k*n)==k), 0) # Chai Wah Wu, Nov 27 2025

Crossrefs

Cf. A000005, A033950, A036763, A036764, A090395 (denominator of d(n)/n), A353011 (indices of "late birds" in A090395).

Sequence in context: A269841 A144551 A174457 * A353011 A110821 A262983

Adjacent sequences: A390910 A390911 A390912 * A390914 A390915 A390916

Keyword

nonn,easy

Author

Joe Anderson, Nov 23 2025

Status

approved