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A382483 a(n) = smallest number k such that at least one of sigma(n) - k and sigma(n) + k is a perfect number.

%I A382483 #27 Apr 08 2025 21:58:29
%S A382483 5,3,2,1,0,6,2,9,7,10,6,0,8,4,4,3,10,11,8,14,4,8,4,32,3,14,12,28,2,44,
%T A382483 4,35,20,26,20,63,10,32,28,62,14,68,16,56,50,44,20,96,29,65,44,70,26,
%U A382483 92,44,92,52,62,32,140,34,68,76,99,56,116,40,98,68,116,44,167,46,86,96,112,68,140
%N A382483 a(n) = smallest number k such that at least one of sigma(n) - k and sigma(n) + k is a perfect number.
%F A382483 a(A081357(k)) = 0.
%F A382483 a(A146542(k)) = 0.
%F A382483 a(A000396(k)) = A000396(k).
%e A382483 sigma(6) = 12, the nearest perfect number is 6, thus a(6) = 12 - 6 = 6.
%e A382483 sigma(26) = 42, the nearest perfect number is 28, thus a(26) = 42 - 28 = 14.
%p A382483 isA000396 := proc(n::integer)
%p A382483     if n < 6 then
%p A382483         false ;
%p A382483     elif numtheory[sigma](n) = 2*n then
%p A382483         true;
%p A382483     else
%p A382483         false;
%p A382483     end if;
%p A382483 end proc:
%p A382483 A382483 := proc(n)
%p A382483     local k ;
%p A382483     for k from 0 do
%p A382483         if isA000396(numtheory[sigma](n)-k) or isA000396(numtheory[sigma](n)+k)  then
%p A382483             return k;
%p A382483         end if;
%p A382483     end do:
%p A382483 end proc:
%p A382483 seq(A382483(n),n=1..50) ; # _R. J. Mathar_, Apr 01 2025
%o A382483 (PARI) isp(x) = if (x>0, sigma(x) == 2*x);
%o A382483 a(n) = my(k=0, s=sigma(n)); while (!(isp(s-k) || isp(s+k)), k++); k; \\ _Michel Marcus_, Apr 01 2025
%Y A382483 Cf. A000396, A081357, A146542, A382506.
%K A382483 nonn,easy
%O A382483 1,1
%A A382483 _Leo Hennig_, Mar 27 2025