A192268 -id:A192268 - cp's OEIS frontend

A192819 Numbers which are both abundant (A005101) and anti-abundant (A192268).

12, 18, 20, 30, 40, 42, 48, 60, 66, 70, 72, 78, 80, 84, 88, 90, 100, 102, 104, 108, 112, 120, 126, 132, 138, 140, 144, 150, 160, 162, 168, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 220, 222, 228, 234, 240, 252, 258, 260, 264, 270, 272, 276, 280, 282, 288

Offset: 1

Jonathan Vos Post, Jul 11 2011

nonn

This is to abundant numbers as A192818 is to deficient numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005101, A066417, A192268, A192818.

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 *)

A005101 INTERSECTION A192268.

Missing term a(4) inserted and more terms added by Amiram Eldar, Jan 13 2022

A192267 Anti-deficient numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 16, 19, 24, 26, 29, 34, 36, 44, 51, 54, 61, 64, 69, 79, 89, 96, 106, 114, 131, 134, 139, 141, 146, 156, 159, 166, 169, 174, 191, 194, 201, 209, 211, 216, 219, 224, 226, 236, 239, 244, 246, 251, 254, 261, 271, 274, 289, 296, 299, 309, 316

Offset: 1

Paolo P. Lava, Jun 28 2011

An anti-deficient number is a number n for which sigma*(n) < n, where sigma*(n) is the sum of the anti-divisors of n. Like A005100 but using anti-divisors. There are only 22 anti-deficient numbers less than 100, 159 less than 1000 and 1547 less than 10000. From an empirical observation it seems that the anti-deficient are approximately less than 18% of the anti-abundant.

			24 is anti-deficient because its anti-divisors are 7, 16 and their sum is 23 < 24.

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

Cf. A066417, A005100, A066272, A192268.

isA192267 := proc(n) A066417(n) < n ; end proc:
for n from 1 to 500 do if isA192267(n) then printf("%d,",n); end if; end do: # R. J. Mathar, Jul 04 2011

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

A000027 = A073930 UNION {this set} UNION A192268.

Edited by Ray Chandler, Dec 05 2011

A192270 Pseudo anti-perfect numbers.

Original entry on oeis.org

5, 7, 8, 10, 17, 22, 23, 31, 32, 33, 35, 38, 39, 41, 45, 49, 52, 53, 56, 59, 60, 63, 67, 68, 70, 71, 72, 73, 74, 76, 77, 81, 82, 83, 85, 88, 94, 95, 98, 101, 102, 103, 104, 105, 108, 109, 110, 112, 115, 116, 117, 122, 123, 127, 129, 130, 137, 138, 143, 144, 147, 148, 149, 150, 151, 154, 157, 158, 162, 164, 165, 167, 171, 172, 175, 176, 178, 179, 182, 185

Offset: 1

Paolo P. Lava, Jun 28 2011

nonn

A pseudo anti-perfect number is a positive integer which is the sum of a subset of its anti-divisors. By definition, anti-perfect numbers (A073930) are a subset of this sequence.

Prime pseudo anti-perfect numbers begin: 5, 7, 17, 23, 31, 41, 53, 59, 67, 71, 73, 83, 101, 103, 109, 127, 137, 149, 151, 157, 167, 179, .... - Jonathan Vos Post, Jul 09 2011

			39 is pseudo anti-perfect because its anti-divisors are 2, 6, 7, 11, 26 and the subset of 2, 11, and 26 adds up to 39.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A005835, A066272, A192268.

with(combinat);
P:=proc(i)
local a,k,n,S;
for n from 1 to i do
  a:={};
  for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a:=a union {k}; fi; od;
  S:=subsets(a);
  while not S[finished] do
    if convert(S[nextvalue](), `+`)=n then print(n); break; fi;
  od;
od;
end:
P(10000);

A192269 Super anti-abundant numbers.

Original entry on oeis.org

1, 3, 4, 5, 7, 13, 17, 32, 38, 45, 67, 77, 143, 203, 247, 473, 682, 787, 1463, 2678, 2992, 3465, 8662, 10868, 16065, 25987, 26163, 29452, 112613, 157658, 202702, 233415, 363825, 795217, 1148647, 1914412, 2139637, 5743237, 5743238, 8393963, 11869357, 64353712

Offset: 1

Paolo P. Lava, Jun 28 2011

nonn

Like A004394 but using anti-divisors. A super anti-abundant number is a number n such that sigma*(n)/n > sigma*(k)/k for all kA066417(n)/n.

			1 -> sigma*(1)/1 = 0/1 = 0;
3 -> sigma*(3)/3 = 2/3 = 0.6666...;
4 -> sigma*(4)/4 = 3/4 = 0.75;
5 -> sigma*(5)/5 = 5/5 = 1;
7 -> sigma*(7)/7 = 10/7 = 1.4285...; etc.

Jud McCranie, Table of n, a(n) for n = 1..66

Cf. A004394, A066417, A192268.

with(numtheory); P:= proc(n) local a,k,i,j,s; s:=0; print(1);
for i from 3 to n do
k:=0; j:=i; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; a:=sigma(2*i+1)+sigma(2*i-1)+sigma(i/2^k)*2^(k+1)-6*i-2;
if a/i>s then s:=a/i; print(i); fi; od; end: P(50000);

a(26)-a(42) from Donovan Johnson, Sep 07 2011

A192287 Quasi-antiperfect numbers.

Original entry on oeis.org

11, 12, 21, 111, 979, 19521, 279259, 4841411, 7231219, 10238379, 14645479, 136531171, 592994139, 1869506239, 13820158011, 35242846899, 211443753471, 330984643659, 8806335754299

Offset: 1

Paolo P. Lava, Aug 02 2011

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

a(16) > 2*10^10. - Donovan Johnson, Sep 22 2011

			Anti-divisors of 979 are 2, 3, 19, 22, 103, 178, 653. Their sum is 980 and 980 = 979+1.

Eric Weisstein's World of Mathematics, Quasiperfect number

Cf. A066272, A073930, A192268, A192288.

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);

sad(n) = vecsum(select(t->n%t && tA066417
isok(n) = sad(n) == n+1; \\ Michel Marcus, Oct 12 2019

a(7)-a(15) from Donovan Johnson, Sep 22 2011
a(16)-a(18) by Jud McCranie, Aug 31 2019
a(19) by Jud McCranie, Oct 10 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 *)

A192271 Anti-weird numbers.

Original entry on oeis.org

11, 12, 13, 14, 15, 18, 20, 21, 25, 27, 28, 30, 37, 40, 42, 43, 46, 47, 48, 50, 55, 57, 58, 62, 65, 66, 75, 78, 80, 84, 86, 87, 90, 91, 92, 93, 97, 99, 100, 107, 111, 113, 118, 119, 120, 121, 124, 125, 126, 128, 132, 133, 135, 136, 140, 142, 145, 152, 153, 155, 160, 161, 163, 168, 170, 173, 177, 180, 181, 183, 184, 186, 188, 190, 192, 196, 197, 198, 204, 205, 208, 210, 212, 213, 218, 222, 223

Offset: 1

Paolo P. Lava, Jun 28 2011

nonn

Like A006037 but using anti-divisors: Anti-weird numbers are anti-abundant (A192268) but not pseudo anti-perfect (A192270).

			25 is an anti-weird number because it is anti-abundant (its anti-divisors are 2, 3, 7, 10, 17 and their sum is 39 > 25) and no subsets of its anti-divisors add up to 25.

Cf. A006037, A066272, A192268, A192270.

# see A066272
isA192270 := proc(n) local a,S ; a := antidivisors(n) ;  S := combinat[subsets](a) ; while not S[finished] do if convert(S[nextvalue](),`+`) = n then return true; end if; end do; false ; end proc:
isA192268 := proc(n) A066417(n) > n ; end proc:
isA192271 := proc(n) isA192268(n) and not isA192270(n) ; end proc:
for n from 1 to 40 do if isA192271(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Jul 04 2011

cp's OEIS Frontend

A192819 Numbers which are both abundant (A005101) and anti-abundant (A192268).

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A192267 Anti-deficient numbers.

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A192270 Pseudo anti-perfect numbers.

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A192269 Super anti-abundant numbers.

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A192287 Quasi-antiperfect 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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A192271 Anti-weird 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.