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.

1, 2, 7, 10, 13, 17, 22, 27, 28, 32, 38, 45, 52, 60, 63, 67, 77, 95, 105, 130, 137, 143, 157, 158, 175, 193, 203, 247, 297, 315, 357, 423, 462, 472, 473, 578, 675, 682, 742, 770, 787, 1012, 1138, 1215, 1417, 1463, 1732, 1957, 2047, 2048, 2327, 2363, 2632

Offset: 1

Author

Paolo P. Lava, Jan 04 2012

Keywords

• nonn

Comments

Anti-imperfect numbers are anti-deficient numbers or anti-abundant numbers.

Examples

			n=1. Anti-divisors: 0. |0-1|=1
n=2. Anti-divisors: 0. |0-2|=2
n=3. Anti-divisors: 2. |2-3|=1 less than 2: 3 is not in the sequence.
n=4. Anti-divisors: 3. |3-4|=1 less than 2: 4 is not in the sequence.
n=5. Anti-divisors: 2,3. |5-3|=2 equal to the maximum: 5 is not in the sequence.
n=6. Anti-divisors: 4. |4-6|=2 equal to the maximum: 6 is not in the sequence.
n=7. Anti-divisors: 2,3,5. |10-7|=3 new maximum: 7 is in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..600 (terms 1..100 from Paolo P. Lava)

Crossrefs

Cf. A066417, A074918, A192267, A192268.

Programs

Maple

P:=proc(i)
local a,k,n,s;
s:=0;
for n from 1 to i do
  a:=0;
  for k from 2 to n-1 do if abs((n mod k)- k/2)<1 then a:=a+k; fi; od;
  if abs(n-a)>s then s:=abs(n-a); print(n); fi;
od;
end:
P(3000);

Mathematica

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 *)