A392361 a(n) is the minimum number of proper divisors of the semiperfect number k = A005835(n) that sum to k.

3, 3, 3, 4, 3, 5, 3, 3, 4, 3, 3, 3, 5, 3, 3, 3, 3, 4, 3, 6, 3, 3, 4, 3, 6, 3, 5, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 3, 6, 3, 3, 3, 5, 3, 4, 3, 6, 3, 3, 4, 3, 5, 3, 3, 3, 3, 3, 3, 4, 3, 3, 6, 3, 4, 3, 3, 3, 3, 7, 3, 5, 3, 3, 4, 3, 3, 3, 4, 3, 3, 7, 6, 3, 3, 5, 3, 8, 3, 3, 4, 3, 3, 5, 3, 4, 3, 3

Offset

1,1

Comments

a(n) <= A390834(n), with equality if and only if A005835(n) is in A064771.

a(n) >= 3, with equality if and only if A005835(n) is a multiple of 6.

a(m) <= a(n) if A005835(m) is a multiple of A005835(n).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

A005835(34) = 144 = 24 + 48 + 72, and {24, 48, 72} is the smallest set of proper divisors of 144 that sums to 144, so a(34) = 3.

Maple

issum:= proc(n, S) option remember; local m, Sp, v; # return a subset of S summing to n
  Sp:= select(`<=`, S, n);
  if n > convert(Sp, `+`) then return false fi;
  if member(n, Sp) then return {n} fi;
  m:= max(Sp); Sp:= Sp minus {m};
  v:= procname(n-m, Sp);
  if v <> false then return v union {m} fi;
  procname(n, Sp)
end proc:
issumle:= proc(n, S, k) option remember; local m, Sp, v;
  if k = 0 then return false fi;
  Sp:= select(`<=`, S, n);
  if n > convert(Sp, `+`) then return false fi;
  if member(n, Sp) then return {n} fi;
  m:= max(Sp); Sp:= Sp minus {m};
  v:= procname(n-m, Sp, k-1);
  if v <> false then return v union {m} fi;
  procname(n, Sp, k)
end proc:
f:= proc(n) local DD, v, k, kp;
    if n mod 6 = 0 then return 3 fi;
    DD:= numtheory:-divisors(n) minus {n};
    v:= issum(n, DD);
    if v = false then return NULL fi;
    k:= nops(v);
    do
      v:= issumle(n, DD, k-1);
      if v = false then return k fi;
      k:= nops(v)
    od;
end proc:
map(f, [$2..500]);

Prog

(Python)
from sympy import proper_divisors
def min_divs(n: int) -> int:
    if n <= 1: return -1
    dp = {0: 0}
    for d in proper_divisors(n):
        n_dp = dp.copy()
        for s, count in dp.items():
            n_sum = s + d
            if n_sum <= n:
                n_count = count + 1
                if n_sum not in n_dp or n_count < n_dp[n_sum]:
                    n_dp[n_sum] = n_count
        dp = n_dp
    return dp.get(n, -1)
print([a for n in range(1, 500) if (a := min_divs(n)) > 0]) # Peter Luschny, Jan 08 2026

Crossrefs

Cf. A005835, A064771, A389712, A390834.

Sequence in context: A198020 A098037 A079108 * A165605 A230194 A128112

Adjacent sequences: A392358 A392359 A392360 * A392362 A392363 A392364

Keyword

nonn

Author

Robert Israel, Jan 08 2026

Status

approved