A073930 -id:A073930 - cp's OEIS frontend

A192268 Anti-abundant numbers.

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}.

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

A214842 Anti-multiply-perfect numbers. Numbers n for which sigma(n)/n is an integer, where sigma(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

1, 2, 5, 8, 41, 56, 77, 946, 1568, 2768, 5186, 6874, 8104, 17386, 27024, 84026, 167786, 2667584, 4775040, 4921776, 27914146, 505235234, 3238952914, 73600829714, 455879783074, 528080296234, 673223621664, 4054397778846, 4437083907194, 4869434608274, 6904301600914, 7738291969456

Offset: 1

Paolo P. Lava, Mar 08 2013

nonn

A073930 and A073931 are subsets of this sequence.
Like A007691 but using sigma*(n) (A066417) instead of sigma(n) (A000203).
Tested up to 167786. Additional terms are 2667584, 4775040, 4921776, 27914146, 505235234, 3238952914, 73600829714 but there may be missing terms among them.

			Anti-divisors of 77 are 2, 3, 5, 9, 14, 17, 22, 31, 51. Their sum is 154 and 154/77=2.

Cf. A000203, A007691, A066272, A066417, A073930, A073931.

A214842:= proc(q) local a,k,n;
for n from 1 to q 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 type(a/n,integer) then print(n); fi; od; end:
A214842(10^10);

a066417[n_Integer] := Total[Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]]; a214842[n_Integer] := Select[Range[n], IntegerQ[a066417[#]/#] &];
a214842[1200] (* Michael De Vlieger, Aug 08 2014 *)

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

A214842 = [n for n in range(1,10**4) if not (sum([d for d in range(2,n,2) if n%d and not 2*n%d])+sum([d for d in range(3,n,2) if n%d and 2*n%d in [d-1,1]])) % n]
# Chai Wah Wu, Aug 12 2014

Verified there are no missing terms up to a(24) by Donovan Johnson, Apr 13 2013
a(25)-a(27) by Jud McCranie, Aug 31 2019
a(28)-a(32) by Jud McCranie, Oct 10 2019

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);

A192290 Anti-amicable numbers.

Original entry on oeis.org

14, 16, 92, 114, 5566, 6596, 1077378, 1529394, 3098834, 3978336, 70774930, 92974314

Offset: 1

Paolo P. Lava, Jun 29 2011

Like A063990 but using anti-divisors. sigma*(a)=b and sigma*(b)=a, where sigma*(n) is the sum of  the anti-divisors of n. Anti-perfect numbers A073930 are not included in the sequence.
There are also chains of 3 or more anti-sociable numbers.
With 3 numbers the first chain is: 1494, 2056, 1856.
sigma*(1494) = 4+7+12+29+36+49+61+103+332+427+996 = 2056.
sigma*(2056) = 3+9+16+1371+457 = 1856.
sigma*(1856) = 3+47+79+128+1237 = 1494.
With 4 numbers the first chain is: 46, 58, 96, 64.
sigma*(46) = 3+4+7+13+31 = 58.
sigma*(58) = 3+4+5+9+13+23+39 = 96.
sigma*(96) = 64.
sigma*(64) = 3+43 = 46.
No other pairs with the larger term < 2147000000. - Jud McCranie Sep 24 2019

			sigma*(14) = 3+4+9 = 16; sigma*(16) = 3+11 = 14.
sigma*(92) = 3+5+8+37+61= 114; sigma*(114) = 4+12+76 = 92.
sigma*(5566) = 3+4+9+44+92+484+1012+1237+3711= 6596; sigma*(6596) = 3+8+79+136+776+167+4397 = 5566.

Cf. A063990, A066272, A192291, A192292, A192293.

with(numtheory);
A192290 := proc(q)
local a,b,c,k,n;
for n from 1 to q 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;
  b:=a; c:=0;
  for k from 2 to b-1 do if abs((b mod k)-k/2)<1 then c:=c+k; fi; od;
  if n=c and not a=c then print(n); fi;
od; end:
A192290(1000000000);

from sympy import divisors
def sigma_s(n):
    return sum([2*d for d in divisors(n) if n > 2*d and n % (2*d)] +
        [d for d in divisors(2*n-1) if n > d >=2 and n % d] +
        [d for d in divisors(2*n+1) if n > d >=2 and n % d])
A192290 = [n for n in range(1,10**4) if sigma_s(n) != n and sigma_s(sigma_s(n)) == n] # Chai Wah Wu, Aug 14 2014

a(7)-a(12) from Donovan Johnson, Sep 12 2011

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

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

A219053 Numbers which are the sum of their proper divisors and their anti-divisors.

Original entry on oeis.org

3, 19, 131, 139, 649, 3649, 4181, 5881, 13369, 29501, 65789, 37548761, 63919409, 2471800109

Offset: 1

Paolo P. Lava, Apr 17 2013

a(15) > 10^10. - Donovan Johnson, Apr 19 2013

			The proper divisors of 649 are 1, 11, 59 and its anti-divisors are 2, 3, 22, 118, 433. Their sum 1+11+59+2+3+22+118+433 is equal to 649.

Cf. A000005, A000396, A001065, A066272, A066417, A073930.

with(numtheory); A219053:=proc(q) local  a,k,n;
for n from 1 to q 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 2*n=sigma(n)+a then print(n); fi; od; end: A219053(10^6);

a(12)-a(14) from Donovan Johnson, Apr 19 2013

A258786 Numbers n whose sum of anti-divisors is a permutation of their digits.

Original entry on oeis.org

5, 8, 41, 56, 64, 358, 614, 946, 1092, 1382, 1683, 2430, 2683, 2734, 2834, 2945, 3045, 3067, 3602, 4056, 4286, 5186, 5784, 6874, 7251, 8104, 8546, 9264, 12881, 14028, 14384, 15258, 17386, 21103, 22044, 23331, 24434, 24603, 25346, 26420, 26822, 26845, 27024, 27232

Offset: 1

Paolo P. Lava, Jun 10 2015

A073930 is a subset of this sequence.

			Anti-divisors of 5 are 2, 3 whose sum is 5.
Anti-divisors of 41 are 2, 3, 9, 27 whose sum is 41.
Anti-divisors of 64 are 3, 43 whose sum is 46 that is a permutation of the digit of 64.

Paolo P. Lava, Table of n, a(n) for n = 1..200

Cf. A066417, A073930, A114065, A115920, A115921, A175795, A225902.

with(numtheory):P:=proc(q) local a,b,j,k,ok,n,p;
for n from 1 to q do k:=0; j:=n;
while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
if ilog10(n)=ilog10(a) then j:=sort(convert(n,base,10)); a:=sort(convert(a,base,10)); ok:=1;
for k from 1 to nops(a) do if j[k]<>a[k] then ok:=0; break;
fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^9);

ad[n_] := Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; Select[Range@ 5000, SameQ[DigitCount@ #, DigitCount[Total[ad@ #]]] &] (* _Michael De Vlieger, Jun 10 2015 *)

from sympy.ntheory.factor_ import antidivisors
A258786_list = [n for n in range(1,10**5) if sorted(str(n)) == sorted(str(sum(antidivisors(n))))] # Chai Wah Wu, Jun 11 2015

A240968 Unitary anti-perfect numbers.

Original entry on oeis.org

5, 8, 10, 41, 206, 1066, 2412, 3281, 8086, 11570, 29525, 57012, 73728, 410390, 413486, 775130, 2391485, 2454146, 2937446, 64563520, 100531166, 152032126, 988747406

Offset: 1

Paolo P. Lava, Aug 05 2014

For any number x we consider the sum of its anti-divisors which are coprime to x (unitary anti-divisors). The sequence list the numbers for which this sum is equal to x.
Subset of A192270.
I found only 2 unitary anti-amicable numbers: 18208, 20470.
No other terms < 2147000000. Jud McCranie, Sep 21 2019.

			Anti-divisors of 1066 are 3, 4, 9, 27, 52, 79, 164, 237, 711. The anti-divisors which are coprime to 1066 are 3, 9, 27, 79, 237, 711 and their sum is 3 + 9 + 27 + 79 + 237 + 711 = 1066.

Cf. A066272, A066417, A073930, A192270, A240979.

P:=proc(q) local a,k,n;
for n from 3 to q do a:=0; b:=0;
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then
  if gcd(n,k)=1 then a:=a+k; fi; fi; od;
if n=a then print(n); fi; od; end: P(10^6);

a(14)-a(23) by Jud McCranie, Sep 21 2019.

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A192268 Anti-abundant numbers.

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A192267 Anti-deficient numbers.

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A214842 Anti-multiply-perfect numbers. Numbers n for which sigma*(n)/n is an integer, where sigma*(n) is the sum of the anti-divisors of n.

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A192270 Pseudo anti-perfect numbers.

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A192290 Anti-amicable numbers.

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A192288 Almost anti-perfect numbers.

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A192287 Quasi-antiperfect numbers.

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A214842 Anti-multiply-perfect numbers. Numbers n for which sigma(n)/n is an integer, where sigma(n) is the sum of the anti-divisors of n.