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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

%I #17 Jan 13 2022 04:23:09

%S 1,2,7,10,13,17,22,27,28,32,38,45,52,60,63,67,77,95,105,130,137,143,

%T 157,158,175,193,203,247,297,315,357,423,462,472,473,578,675,682,742,

%U 770,787,1012,1138,1215,1417,1463,1732,1957,2047,2048,2327,2363,2632

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

%H 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 n=1. Anti-divisors: 0. |0-1|=1

%e n=2. Anti-divisors: 0. |0-2|=2

%e n=3. Anti-divisors: 2. |2-3|=1 less than 2: 3 is not in the sequence.

%e n=4. Anti-divisors: 3. |3-4|=1 less than 2: 4 is not in the sequence.

%e n=5. Anti-divisors: 2,3. |5-3|=2 equal to the maximum: 5 is not in the sequence.

%e n=6. Anti-divisors: 4. |4-6|=2 equal to the maximum: 6 is not in the sequence.

%e n=7. Anti-divisors: 2,3,5. |10-7|=3 new maximum: 7 is in the sequence.

%p P:=proc(i)

%p local a,k,n,s;

%p s:=0;

%p for n from 1 to i do

%p a:=0;

%p for k from 2 to n-1 do if abs((n mod k)- k/2)<1 then a:=a+k; fi; od;

%p if abs(n-a)>s then s:=abs(n-a); print(n); fi;

%p od;

%p end:

%p P(3000);

%t 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 Cf. A066417, A074918, A192267, A192268.

%K nonn

%O 1,2

%A _Paolo P. Lava_, Jan 04 2012

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Last modified March 29 07:27 EDT 2024. Contains 371265 sequences. (Running on oeis4.)