A375389 - OEIS

%I #15 Aug 17 2024 22:28:25

%S -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,

%T 12,-1,-1,-1,-1,-1,12,-1,12,-1,-1,-1,12,-1,18,-1,20,-1,12,-1,20,-1,-1,

%U -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

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

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

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

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

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

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

%H Robert Israel, <a href="/A375389/b375389.txt">Table of n, a(n) for n = 1..21000</a>

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

%p Ab:= select(t -> numtheory:-sigma(t) > 2*t, [$1..10^4]):

%p f:= proc(n) local i,x;

%p   for i from 1 do

%p     x:= Ab[i];

%p     if 2*x > n then return -1 fi;

%p     if ListTools:-BinarySearch(Ab, n-x) <> 0 then return x fi

%p   od;

%p end proc:

%p map(f, [$1..100]);

%Y Cf. A005101, A048242.

%K sign,look

%O 1,24

%A _Robert Israel_, Aug 13 2024