A393185 - OEIS

%I #7 Feb 12 2026 17:22:13

%S 5,3,13,6,25,11,29,22,58,87,61,84,112,73,108,59,156,46,50,97,19,157,

%T 214,79,192,142,193,181,27,269,141,243,312,280,158,94,53,229,109,106,

%U 180,358,252,89,384,47,406,120,277,446,127,433,17,337,261,250,101,421

%N Numbers that are the smaller of two distinct positive integers with opposite abundances, where this pair of abundances occurs for the first time; sorted by the larger term.

%C The larger positive integer is A393184(n), the deficiency of a(n) is A393186(n).

%C For each n, the pair (A393185(n), A393184(n)) contains one abundant number and one deficient number. Exception: at n = 4, the pair (6, 28) consists of two perfect numbers.

%H Felix Huber, <a href="/A393185/b393185.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Abundance.html">Abundance</a>

%e a(1) = 5 since (5, A393184(1)) = (5, 12) is the first pair with opposite abundances +-4.

%e a(4) = 6 since (6, A393184(4)) = (6, 28) is the first pair with abundances 0 and 0.

%e a(12) = 112 since (112, A393184(12)) = (112, 118) is the first pair with opposite abundances +-56.

%p A393185List := proc(N)

%p     option remember;

%p     local a, t, s, k, d, i;

%p     a := `if`(N < 28, [], [[28, 6]]);

%p     t := table();

%p     s := table();

%p     for k to N do

%p         d := NumberTheory:-sigma(k) - 2*k;

%p         if not assigned(t[d]) then

%p             t[d] := k;

%p             if assigned(t[-d]) then

%p                 if not assigned(s[k]) then

%p                     a := [op(a), [k, min(k, t[-d])]];

%p                     s[k] := 1;

%p                 end if;

%p             end if;

%p         end if;

%p     end do;

%p     a := sort(a);

%p     seq(a[i][2], i = 2 .. nops(a));

%p end proc:

%p A393185List(588);

%Y Cf. A000203, A000396, A002025, A005100, A005101, A033879, A033880, A330901, A331181, A335253, A335254, A393184, A393186.

%K nonn,easy

%O 1,1

%A _Felix Huber_, Feb 08 2026