A392648 Numbers k such that there exist at least two distinct subsets s1 and s2 of {1 < d < k: d | k}, such that Sum(s1) = Sum(s2) = k.

24, 30, 36, 48, 54, 60, 66, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 180, 192, 198, 200, 204, 210, 216, 220, 224, 228, 234, 240, 252, 260, 264, 270, 276, 280, 288, 294, 300, 306, 308, 312, 320, 324, 330, 336, 340, 342, 348

Offset

1,1

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Example

Witnesses for the first few terms:
  24 -> [[2, 4, 6, 12], [4, 8, 12]]
  30 -> [[2, 3, 10, 15], [5, 10, 15]]
  36 -> [[2, 3, 4, 6, 9, 12], [2, 3, 4, 9, 18]]
  48 -> [[2, 4, 6, 8, 12, 16], [2, 4, 6, 12, 24]]
  54 -> [[3, 6, 18, 27], [9, 18, 27]]
  60 -> [[2, 3, 4, 5, 6, 10, 30], [2, 3, 4, 6, 10, 15, 20]]
  66 -> [[2, 3, 6, 22, 33], [11, 22, 33]]
  72 -> [[2, 3, 4, 6, 9, 12, 36], [2, 3, 4, 9, 12, 18, 24]]
  80 -> [[2, 4, 8, 10, 16, 40], [2, 8, 10, 20, 40]]
  84 -> [[2, 3, 4, 7, 12, 14, 42], [2, 3, 4, 12, 14, 21, 28]]
  90 -> [[2, 3, 6, 9, 10, 15, 45], [2, 3, 10, 30, 45]]
  96 -> [[2, 4, 6, 8, 12, 16, 48], [2, 4, 6, 12, 16, 24, 32]]
The first odd term is 945 -> [[3, 7, 21, 27, 35, 45, 63, 105, 135, 189, 315], [7, 9, 15, 27, 35, 45, 63, 105, 135, 189, 315]].

Maple

q:= proc(n) option remember; local b, l, ll:
      l, b:= sort([(numtheory[divisors](n) minus {1, n})[]]),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1 or ll[i]<m, 0,
        (t-> t+`if`(t>1, 0, `if`(l[i]>m, 0, b(m-l[i], i-1))))(b(m, i-1))))
      end; ll:= ListTools[PartialSums](l); is(b(n, nops(l))>1)
    end:
select(q, [$1..1000])[];  # Alois P. Heinz, Jan 22 2026

Prog

(Python)
from sympy import divisors
def isA392648(n: int) -> int:
    if n == 0: return False
    divs = divisors(n)[1:-1]
    dp = [0] * (n + 1)
    dp[0] = 1
    for d in divs:
        for j in range(n, d - 1, -1):
            dp[j] += dp[j - d]
    return dp[n] > 1
print([n for n in range(350) if isA392648(n)])
(PARI) isok(k) = my(d=setminus(divisors(k), Set([1, k]))); forsubset(#d, s, my(x=sum(i=1, #s, d[s[i]])); if (x==k, forsubset(#d, t, if((s!=t) && (sum(i=1, #t, d[t[i]]) == k), return(1))))); \\ Michel Marcus, Jan 22 2026

Crossrefs

Cf. A070824, A136446.

Sequence in context: A230195 A098030 A290451 * A068544 A284174 A292982

Adjacent sequences: A392645 A392646 A392647 * A392649 A392650 A392651

Keyword

nonn

Author

Peter Luschny, Jan 18 2026

Status

approved