cp's OEIS Frontend

A203621 Highly anti-imperfect numbers: numbers k that sets a record for the value of |sigma*(k)-k|, where sigma*(k) is the sum of the anti-divisors of k.

%I A203621 #17 Jan 13 2022 04:23:09
%S A203621 1,2,7,10,13,17,22,27,28,32,38,45,52,60,63,67,77,95,105,130,137,143,
%T A203621 157,158,175,193,203,247,297,315,357,423,462,472,473,578,675,682,742,
%U A203621 770,787,1012,1138,1215,1417,1463,1732,1957,2047,2048,2327,2363,2632
%N A203621 Highly anti-imperfect numbers: numbers k that sets a record for the value of |sigma*(k)-k|, where sigma*(k) is the sum of the anti-divisors of k.
%C A203621 Anti-imperfect numbers are anti-deficient numbers or anti-abundant numbers.
%H A203621 Amiram Eldar, <a href="/A203621/b203621.txt">Table of n, a(n) for n = 1..600</a> (terms 1..100 from Paolo P. Lava)
%e A203621 n=1. Anti-divisors: 0. |0-1|=1
%e A203621 n=2. Anti-divisors: 0. |0-2|=2
%e A203621 n=3. Anti-divisors: 2. |2-3|=1 less than 2: 3 is not in the sequence.
%e A203621 n=4. Anti-divisors: 3. |3-4|=1 less than 2: 4 is not in the sequence.
%e A203621 n=5. Anti-divisors: 2,3. |5-3|=2 equal to the maximum: 5 is not in the sequence.
%e A203621 n=6. Anti-divisors: 4. |4-6|=2 equal to the maximum: 6 is not in the sequence.
%e A203621 n=7. Anti-divisors: 2,3,5. |10-7|=3 new maximum: 7 is in the sequence.
%p A203621 P:=proc(i)
%p A203621 local a,k,n,s;
%p A203621 s:=0;
%p A203621 for n from 1 to i do
%p A203621   a:=0;
%p A203621   for k from 2 to n-1 do if abs((n mod k)- k/2)<1 then a:=a+k; fi; od;
%p A203621   if abs(n-a)>s then s:=abs(n-a); print(n); fi;
%p A203621 od;
%p A203621 end:
%p A203621 P(3000);
%t A203621 sig[n_] := Total[Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]]; d[n_] := Abs[sig[n] - n]; s = {}; dm = -1; Do[If[(d1 = d[n]) > dm, dm = d1; AppendTo[s, n]], {n, 1, 2700}]; s (* _Amiram Eldar_, Jan 13 2022 after _Michael De Vlieger_ at A066417 *)
%Y A203621 Cf. A066417, A074918, A192267, A192268.
%K A203621 nonn
%O A203621 1,2
%A A203621 _Paolo P. Lava_, Jan 04 2012