A392648 - OEIS

%I #17 Jan 22 2026 08:14:54

%S 24,30,36,48,54,60,66,72,80,84,90,96,108,112,120,126,132,140,144,150,

%T 156,160,162,168,180,192,198,200,204,210,216,220,224,228,234,240,252,

%U 260,264,270,276,280,288,294,300,306,308,312,320,324,330,336,340,342,348

%N 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.

%H Alois P. Heinz, <a href="/A392648/b392648.txt">Table of n, a(n) for n = 1..20000</a>

%e Witnesses for the first few terms:

%e   24 -> [[2, 4, 6, 12], [4, 8, 12]]

%e   30 -> [[2, 3, 10, 15], [5, 10, 15]]

%e   36 -> [[2, 3, 4, 6, 9, 12], [2, 3, 4, 9, 18]]

%e   48 -> [[2, 4, 6, 8, 12, 16], [2, 4, 6, 12, 24]]

%e   54 -> [[3, 6, 18, 27], [9, 18, 27]]

%e   60 -> [[2, 3, 4, 5, 6, 10, 30], [2, 3, 4, 6, 10, 15, 20]]

%e   66 -> [[2, 3, 6, 22, 33], [11, 22, 33]]

%e   72 -> [[2, 3, 4, 6, 9, 12, 36], [2, 3, 4, 9, 12, 18, 24]]

%e   80 -> [[2, 4, 8, 10, 16, 40], [2, 8, 10, 20, 40]]

%e   84 -> [[2, 3, 4, 7, 12, 14, 42], [2, 3, 4, 12, 14, 21, 28]]

%e   90 -> [[2, 3, 6, 9, 10, 15, 45], [2, 3, 10, 30, 45]]

%e   96 -> [[2, 4, 6, 8, 12, 16, 48], [2, 4, 6, 12, 16, 24, 32]]

%e 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]].

%p q:= proc(n) option remember; local b, l, ll:

%p       l, b:= sort([(numtheory[divisors](n) minus {1, n})[]]),

%p       proc(m, i) option remember; `if`(m=0, 1, `if`(i<1 or ll[i]<m, 0,

%p         (t-> t+`if`(t>1, 0, `if`(l[i]>m, 0, b(m-l[i], i-1))))(b(m, i-1))))

%p       end; ll:= ListTools[PartialSums](l); is(b(n, nops(l))>1)

%p     end:

%p select(q, [$1..1000])[];  # _Alois P. Heinz_, Jan 22 2026

%o (Python)

%o from sympy import divisors

%o def isA392648(n: int) -> int:

%o     if n == 0: return False

%o     divs = divisors(n)[1:-1]

%o     dp = [0] * (n + 1)

%o     dp[0] = 1

%o     for d in divs:

%o         for j in range(n, d - 1, -1):

%o             dp[j] += dp[j - d]

%o     return dp[n] > 1

%o print([n for n in range(350) if isA392648(n)])

%o (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

%Y Cf. A070824, A136446.

%K nonn

%O 1,1

%A _Peter Luschny_, Jan 18 2026