A389712 a(n) is the least number k such that there is a set of n proper divisors of k whose sum is k, and no set of fewer than n proper divisors of k whose sum is k.

6, 20, 28, 88, 304, 368, 496, 650, 945, 3230, 2205, 11096, 12705, 20482, 22275, 51850, 7425, 107198, 8925, 126854, 51765, 253946, 46035, 592670, 81081, 133042, 223839, 956650, 78975, 1733750, 279279, 1715098, 474045, 1768102, 1423575

Offset

3,1

Comments

It seems this is a subsequence of Zumkeller numbers (A083207). - Ivan N. Ianakiev, Jan 21 2026

a(39) = 371925. a(38) and a(40) > 5000000 if they exist. - Robert Israel, Jan 21 2026

From David A. Corneth, Jan 25 2026, Jan 30 2026: (Start)

a(n) has at least n divisors; sigma(a(n)) >= 2*a(n).

In data so far sigma(a(n)) <= 2.1 * a(n).

a(m) is no multiple of a(n) for m>n.

In general if k can be written as a sum of t distinct proper divisors of k then a(n) is no larger multiple of k for n > t.

a(38) <= 8043652, a(40) <= 7296842. (End)

Links

Table of n, a(n) for n=3..37.

Robert Israel, Maple code

Formula

a(n) = A005835(k) where A392361(k) = n and A392361(j) <> n for j < k.

Example

a(6) = 88 because 88 = 1 + 2 + 8 + 11 + 22 + 44 is the sum of 6 proper divisors of 88, and not the sum of fewer than 6 proper divisors of 88, and no number smaller than 88 works.
176 is not in the sequence as a proper divisor of 176 (88) is in the sequence. - David A. Corneth, Jan 26 2026

Maple

# See Links

Crossrefs

Cf. A005835, A083207, A392361.

Sequence in context: A308710 A376874 A388030 * A388266 A388272 A242341

Adjacent sequences: A389709 A389710 A389711 * A389713 A389714 A389715

Keyword

nonn,more

Author

Robert Israel, Jan 08 2026

Status

approved