A192267 -id:A192267 - cp's OEIS frontend

A192818 Numbers which are both deficient (A005100) and anti-deficient (A192267).

1, 2, 3, 4, 9, 16, 19, 26, 29, 34, 44, 51, 61, 64, 69, 79, 89, 106, 131, 134, 139, 141, 146, 159, 166, 169, 191, 194, 201, 209, 211, 219, 226, 236, 239, 244, 251, 254, 261, 271, 274, 289, 296, 299, 309, 316, 321, 334, 339, 341, 344, 349, 359, 376, 381, 386

Offset: 1

Jonathan Vos Post, Jul 10 2011

			24 is anti-deficient because its anti-divisors are 7, 16 and their sum is 23 < 24.  26 is deficient because its proper divisors are 1, 2, 13 which sum to 16 and 16 < 26.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1500 from Nathaniel Johnston)

Cf. A005100, A066417, A192267.

q[n_] := Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]] < n && DivisorSigma[1, n] < 2*n; Select[Range[300], q] (* _Amiram Eldar, Jan 13 2022 after Michael De Vlieger at A066417 *)

A005100 INTERSECTION A192267.

More terms and inserted a(1)=1 from Nathaniel Johnston, Sep 26 2011

A192268 Anti-abundant numbers.

Original entry on oeis.org

7, 10, 11, 12, 13, 14, 15, 17, 18, 20, 21, 22, 23, 25, 27, 28, 30, 31, 32, 33, 35, 37, 38, 39, 40, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 55, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113

Offset: 1

Paolo P. Lava, Jun 28 2011

An anti-abundant number is a number n for which sigma*(n) > n, where sigma*(n) is the sum of the anti-divisors of n. Like A005101 but using anti-divisors.

			25 is anti-abundant because its anti-divisors are 2, 3, 7, 10, 17 and their sum is 39 > 25.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Paolo P. Lava)

Cf. A066417, A005101, A066272, A192267.

isA192268 := proc(n) A066417(n) > n ; end proc:
for n from 1 to 500 do if isA192268(n) then printf("%d,",n); end if; end do: # R. J. Mathar, Jul 04 2011

antiAbQ[n_] := Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]] > n; Select[Range[120], antiAbQ] (* _Amiram Eldar, Jan 13 2022 after Michael De Vlieger at A066417 *)

from itertools import count, islice
from sympy import divisor_sigma, multiplicity
def A192268gen(): return filter(lambda n:divisor_sigma(2*n-1)+divisor_sigma(2*n+1)+divisor_sigma(n//2**(k:=multiplicity(2,n)))*2**(k+1)-7*n-2 > 0,count(2))
A192268_list = list(islice(A192268gen(),16)) # Chai Wah Wu, Dec 23 2021

A000027 = A073930 UNION A192267 UNION {this set}.

A192288 Almost anti-perfect numbers.

Original entry on oeis.org

3, 4, 9, 19, 24, 131, 139, 339, 5881, 14849, 29501, 57169, 63061, 65789, 542781, 2439241, 3197249, 4111561, 8614481, 48657789, 218234169, 309296261, 731499089, 1191549689, 1569571661, 2471800109, 5687426561, 9505043161, 67784277581, 79468538969, 257067141569, 290324629889, 397393221689, 445568135041, 2260763053809

Offset: 1

Paolo P. Lava, Aug 02 2011

nonn

An almost anti-perfect number is a least anti-deficient number, i.e., one such that sigma*(n)=n-1, where sigma*(n) is the sum of the anti-divisors of n. Like almost perfect numbers (see link) but using anti-divisors.

a(29) > 2*10^10. - Donovan Johnson, Sep 22 2011

			Anti-divisors of 5881 are 2, 3, 9, 19, 619, 1307, 3921. Their sum is 5880 and 5880=5881-1.

Jud McCranie, Table of n, a(n) for n = 1..36
Eric Weisstein's World of Mathematics, Almost perfect number

Cf. A066272, A073930, A192267, A192287.

P:=proc(n)
local a,i,k;
for i from 3 to n do
  a:=0;
  for k from 2 to i-1 do
    if abs((i mod k)-k/2)<1 then a:=a+k; fi;
  od;
  if i-1=a then print(i); fi;
od;
end:
P(1000000);

a(15)-a(28) from Donovan Johnson, Sep 22 2011
a(29)-a(34) from Jud McCranie, Aug 31 2019
a(35) from Jud McCranie, Sep 05 2019

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.

Original entry on oeis.org

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

Paolo P. Lava, Jan 04 2012

nonn

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

			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.

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

Cf. A066417, A074918, A192267, A192268.

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);

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 *)

cp's OEIS Frontend

A192818 Numbers which are both deficient (A005100) and anti-deficient (A192267).

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A192268 Anti-abundant numbers.

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A192288 Almost anti-perfect numbers.

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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.

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Maple

Mathematica

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.