A375389 a(n) is the smallest abundant number k such that n - k is abundant, or -1 if there is no such k.

-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 12, -1, -1, -1, -1, -1, 12, -1, 12, -1, -1, -1, 12, -1, 18, -1, 20, -1, 12, -1, 20, -1, -1, -1, 12, -1, 20, -1, 12, -1, 12, -1, 20, -1, 18, -1, 12, -1, 20, -1, 24, -1, 12, -1, 12, -1, 30, -1, 12, -1, 18

Offset: 1

Author

Robert Israel, Aug 13 2024

Keywords

• sign
• look

Comments

a(n) >= 12 for n >= 20162.

a(n) = 12 if n >= 24 and n == 0 (mod 6).

12 <= a(n) <= 20 if n >= 26 and n == 2 (mod 6).

12 <= a(n) <= 40 if n >= 52 and n == 4 (mod 6).

If a(n) > 0 then 0 < a(k n - (k-1) a(n)) <= a(n) for all positive integers k.

Examples

			a(30) = 12 because 30 = 12 + 18 where 12 and 18 are abundant numbers.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..21000

Crossrefs

Cf. A005101, A048242.

Programs

Maple

Ab:= select(t -> numtheory:-sigma(t) > 2*t, [$1..10^4]):
f:= proc(n) local i,x;
  for i from 1 do
    x:= Ab[i];
    if 2*x > n then return -1 fi;
    if ListTools:-BinarySearch(Ab, n-x) <> 0 then return x fi
  od;
end proc:
map(f, [$1..100]);