A066272 -id:A066272 - 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}.

A058838 a(n) = 1 + sum of the anti-divisors of n.

Original entry on oeis.org

1, 1, 3, 4, 6, 5, 11, 9, 9, 15, 13, 14, 20, 17, 19, 15, 29, 29, 19, 25, 23, 37, 35, 24, 40, 25, 43, 47, 25, 37, 43, 59, 49, 31, 53, 33, 51, 71, 53, 56, 42, 67, 57, 41, 87, 59, 61, 57, 73, 81, 43, 95, 89, 53, 75, 57, 75, 97, 91, 108, 58, 79, 113, 47, 85

Offset: 1

Jon Perry, Dec 28 2001

nonn

See A066272 for definition of anti-divisor.

			Consider n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {5,7,35}, {3,9}, {37} respectively and quotients {7, 5, 1}, {12, 4}, {1}; so the anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 1 + 28 = 29.

Jon Perry, Anti-divisors [Broken link]
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A066417, A066241, A066452.

a(n) = A066417(n) + 1.

A066452 Anti-phi(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 4, 3, 2, 8, 3, 7, 7, 9, 2, 8, 5, 10, 10, 8, 6, 19, 6, 12, 9, 9, 8, 22, 9, 12, 12, 15, 10, 31, 9, 11, 14, 24, 13, 23, 9, 24, 17, 16, 10, 35, 15, 23, 25, 20, 12, 40, 17, 34, 21, 18, 14, 37, 17, 24, 25, 41, 20, 39, 14, 31, 34, 33, 18, 42, 16, 32, 37, 41, 18, 44, 25

Offset: 2

Jon Perry, Dec 29 2001

anti-phi(n) = the number of integers < n that are not divisible by any anti-divisor of n.
The old definition given for this sequence was: anti-phi(n) = number of integers <= n that are coprime to the anti-divisors of n. However this does not match the entries.
See A066272 for definition of anti-divisor.

			10 has anti-divisors 3,4,7. The numbers not divisible by any of 3,4,7 and less than 10 are 1,2,5. Therefore anti-phi(10)=3.

Nathaniel Johnston, Table of n, a(n) for n = 2..10000 [This replaces an earlier b-file computed by Diana Mecum]
Jon Perry, Anti-phi function [Broken link]
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A058838, A066241.

# needs antidivisors() as implemented in A066272
A066452 := proc(n)local ad,isad,j,k,totad:ad:=antidivisors(n):totad:=0:for j from 1 to n-1 do isad:=1:for k from 1 to nops(ad) do if(j mod ad[k]=0)then isad:=0:break: fi:od:totad:=totad+isad:od:return totad:end:
seq(A066452(n), n=2..50); # Nathaniel Johnston, Apr 20 2011

antidiv(n) = {my(v = []); for (k=2, n-1, if (abs((n % k) - k/2) < 1, v = concat(v, k));); v;}
a(n) = {my(vad = antidiv(n)); my(nbad = 0); for (j=1, n-1, isad = 1; for (k=1, #vad, if ((j % vad[k]) == 0, isad = 0; break); ); nbad += isad;); nbad;} \\ Michel Marcus, Feb 25 2016

def A066452(n):
    return len([x for x in range(1,n) if all([x % d  for d in range(2,n) if (n % d) and (2*n) % d in [d-1,0,1]])]) # Chai Wah Wu, Aug 07 2014

Better definition and more terms from Diana L. Mecum, Jul 01 2007

A066466 Numbers having just one anti-divisor.

Original entry on oeis.org

3, 4, 6, 96, 393216

Offset: 1

Robert G. Wilson v, Jan 02 2002

See A066272 for definition of anti-divisor.
Jon Perry calls these anti-primes.
A066272(a(n)) = 1.
From Max Alekseyev, Jul 23 2007; updated Jun 25 2025: (Start)
Except for a(2) = 4, the terms of A066466 have form 2^k*p where p is odd prime and both 2^(k+1)*p-1, 2^(k+1)*p+1 are prime (i.e., twin primes). In other words, this sequence, omitting 4, is a subsequence of A040040 containing elements of the form 2^k*p with prime p.
Furthermore, since 2^(k+1)*p-1, 2^(k+1)*p+1 must equal -1 and +1 modulo 3, the number 2^(k+1)*p must be 0 modulo 3, implying that p=3. Therefore every term, except 4, must be of the form 3*2^k such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. In other words, k+1 belongs to the intersection of A002253 and A002235.
According to Ballinger and Keller's lists, there are no other such k up to 22*10^6. Therefore a(6) (if it exists) is greater than 3*2^(22*10^6) ~= 10^6622660. (End)
From Daniel Forgues, Nov 23 2009: (Start)
The 2 last known anti-primes seem to relate to the Fermat primes (coincidence?):
96 = 3 * 2^5 = 3 * 2^F_1 = 3 * 2^[2^(2^1) + 1] and
393216 = 3 * 2^17 = 3 * 2^F_2 = 3 * 2^[2^(2^2) + 1],
where F_k is the k-th Fermat prime. (End)

Ray Ballinger and Wilfrid Keller, List of primes k·2^n + 1 for k < 300
Wilfrid Keller, List of primes k·2^n - 1 for k < 300
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A000215, A002253, A002235, A066272, A019434.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n & ]; Select[ Range[10^5], Length[ antid[ # ]] == 1 & ]

Edited by Max Alekseyev, Oct 13 2009

A178029 Numbers whose sum of divisors equals the sum of their anti-divisors.

Original entry on oeis.org

11, 22, 33, 65, 82, 117, 218, 483, 508, 537, 6430, 21541, 117818, 3589646, 7231219, 8515767, 13050345, 47245905, 50414595, 104335023, 217728002, 1217532421, 1573368218, 1875543429, 2269058065, 11902221245, 12196454655, 12658724029

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, May 19 2010

nonn

			6430 is in the sequence because the sum of divisors is 1+2+5+10+643+1286+3215+6430 = 11592
which equals the sum of anti-divisors 3+4+7+9+11+20+77+167+1169+1429+1837+2572+4287 = 11592.
21541 is in the sequence because the sum of divisors is 1+13+1657+21541 = 23212
and equals the sum of anti-divisors 2+3+9+26+67+643+3314+4787+14361 = 23212.

Cf. A000203, A066272, A066417.

with(numtheory): P:=proc(q) local j,k; k:=0; j:=q; while j mod 2<>1 do k:=k+1; j:=j/2; od; if sigma(q)=sigma(2*q+1)+sigma(2*q-1)+sigma(q/2^k)*2^(k+1)-6*q-2 then q; fi; end: seq(P(i),i=3..10^5);
# alternative Maple implementation:
antidivisors := proc(n) local a,k; a := {} ; for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a := a union {k} ; end if; end do: a ; end proc:
A066417 := proc(n) add(d,d=antidivisors(n)) ; end proc:
isA178029 := proc(n) numtheory[sigma](n) = A066417(n) ; end proc:
for n from 1 do if isA178029(n) then printf("%d,\n",n) ; end if; end do:
# R. J. Mathar, May 24 2010

antidivisors[n_] := Select[Range[2, n-1], Abs[Mod[n, #] - #/2] < 1&];
For[k = 1, k <= 10^5, k++, If[DivisorSigma[1, k] == Total[antidivisors[k]], Print[k]]] (* Jean-François Alcover, Jun 14 2023 *)

from sympy import divisors
[n for n in range(1,10**5) if sum([d for d in range(2,n) if (n % d) and (2*n) % d in [d-1,0,1]]) == sum(divisors(n))] # Chai Wah Wu, Aug 07 2014

{n: A066417(n) = A000203(n)}. - R. J. Mathar, May 24 2010

a(13)-a(28) from Donovan Johnson, Jun 12 2010

A192293 Let sigma_m (n) be the result of applying the sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma_m (n) = k*n; this sequence gives the (2,3)-anti-perfect numbers.

Original entry on oeis.org

32, 98, 2524, 199282, 1336968

Offset: 1

Paolo P. Lava, Jun 29 2011

Like A019281 but using anti-divisors.

a(6) > 2*10^7. - Chai Wah Wu, Dec 02 2014

			sigma*(32)= 3+5+7+9+13+21=58; sigma*(58)= 3+4+5+9+13+23+39=96 and 3*32=96.
sigma*(98)= 3+4+5+13+15+28+39+65=172; sigma*(172)= 3+5+7+8+15+23+49+69+115=294 and 3*98=294.
sigma*(2524)= 3+7+8+9+11+17+27+33+49+51+99+103+153+187+297+459+561+721+1683=4478; sigma*(4478)= 3+4+5+9+13+15+45+53+169+199+597+689+995+1791+2985=7572 and 3*2524=7572.

Cf. A019281, A066272, A192290, A192291, A192292.

with(numtheory): P:= proc(n) local i,j,k,s,s1; for i from 3 to n do
k:=0; j:=i; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; s:=sigma(2*i+1)+sigma(2*i-1)+sigma(i/2^k)*2^(k+1)-6*i-2;
k:=0; j:=s; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; s1:=sigma(2*s+1)+sigma(2*s-1)+sigma(s/2^k)*2^(k+1)-6*s-2;
if s1/i=3 then print(i); fi; od; end: P(10^9);

from sympy import divisors
def antidivisors(n):
    return [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]
A192293_list = []
for n in range(1,10**4):
    if 3*n == sum(antidivisors(sum(antidivisors(n)))):
         A192293_list.append(n) # Chai Wah Wu, Dec 02 2014

a(4)-a(5) from Chai Wah Wu, Dec 01 2014

A066241 1 + number of anti-divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 3, 4, 4, 3, 5, 4, 4, 3, 6, 5, 4, 4, 4, 6, 6, 3, 6, 4, 6, 6, 4, 4, 6, 7, 6, 4, 6, 3, 6, 8, 6, 5, 5, 6, 6, 4, 8, 6, 6, 4, 7, 7, 4, 8, 8, 4, 6, 4, 6, 8, 8, 7, 5, 6, 8, 3, 6, 6, 10, 8, 4, 6, 6, 7, 8, 6, 6, 6, 10, 6, 4, 6, 7, 8, 8, 5, 9, 6, 8, 8, 4, 6, 6, 6, 8, 10, 10, 2, 8, 9, 6, 5

Offset: 1

N. J. A. Sloane, Dec 31 2001

See A066272 for definition of anti-divisor.

			For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {3,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 12 are 4, 5, 7, 12. Therefore a(18) = 1 + 4 = 5.

Jon Perry, Anti-divisors
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A058838. Equals 1 + A066272(n).

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 &], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 &], 2n/Select[ Divisors[2*n], OddQ[ # ] && # != 1 &]]], # < n &]; Table[ Length[ antid[n]] + 1, {n, 1, 100} ]

More terms from Robert G. Wilson v, Jan 03 2002

A091507 Product of the anti-divisors of n.

Original entry on oeis.org

2, 3, 6, 4, 30, 15, 12, 84, 42, 40, 270, 108, 120, 33, 2310, 1680, 78, 312, 168, 8100, 4050, 112, 7140, 204, 11880, 25080, 114, 960, 7938, 257985, 17160, 276, 19320, 192, 11250, 1732500, 24024, 11664, 1458, 114240, 14790, 696, 5896800, 33852, 17670

Offset: 3

Lior Manor, Mar 03 2004

nonn

See A066272 for definition of anti-divisor.

			For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {3,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 12 are 4, 5, 7, 12. Therefore a(18) = 4*5*7*12 = 1680.

Paolo P. Lava, Table of n, a(n) for n = 3..1000
Jon Perry, Anti-divisors.
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A066417.

A091507 := proc(n)
        mul( a, a=antidivisors(n)) ; # reuse A066272
end proc:
seq(A091507(n),n=3..10) ; # R. J. Mathar, Jan 24 2022

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n &]; Table[ Times @@ antid[n], {n, 3, 50}] (* Robert G. Wilson v, Mar 15 2004 *)
a091507[n_Integer] := Apply[Times, Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]]; Array[a091507, 10000] (* _Michael De Vlieger, Aug 08 2014, after Harvey P. Dale at A066272 *)

from functools import reduce
from operator import mul
def A091507(n):
    return reduce(mul,[d for d in range(2,n) if n%d and 2*n%d in [d-1,0,1]]) # Chai Wah Wu, Aug 08 2014

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

cp's OEIS Frontend

A192268 Anti-abundant numbers.

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A058838 a(n) = 1 + sum of the anti-divisors of n.

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A066452 Anti-phi(n).

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A066466 Numbers having just one anti-divisor.

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A178029 Numbers whose sum of divisors equals the sum of their anti-divisors.

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A192293 Let sigma*_m (n) be the result of applying the sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma*_m (n) = k*n; this sequence gives the (2,3)-anti-perfect numbers.

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A066241 1 + number of anti-divisors of n.

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A192293 Let sigma_m (n) be the result of applying the sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma_m (n) = k*n; this sequence gives the (2,3)-anti-perfect numbers.

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.