A066417 -id:A066417 - cp's OEIS frontend

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

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.

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

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

A066418 Numbers k for which phi(k) + anti-phi(k) = k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 12, 15, 27, 30, 40, 44, 57, 117, 128, 171, 236, 399, 408, 510, 1623, 3597, 3915, 4616, 4684, 7335, 10197, 10768, 14144, 32768, 39387, 76035, 77097, 106605, 162450, 196080, 219966, 391696

Offset: 1

Jon Perry, Dec 28 2001

Anti-phi(n) (A066452) is the number of numbers coprime to all the anti-divisors of n.

See A066272 for definition of anti-divisor.

			The anti-divisors of 7 are 1, 2, 3 and 5. Therefore of the integer 1-6, only 1 is coprime to 2, 3 and 5, therefore anti-phi(7)=1. phi(7)=6, therefore anti-phi(7)+phi(7)=7

Jon Perry, Anti-phi function [Wayback Machine link]
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A066416, A066417, A058838, A066241, A066272, A066452.

a(21)-a(34) from Nathaniel Johnston, Apr 20 2011

a(35)-a(39) from Amiram Eldar, Jan 12 2020

A073956 Palindromes whose sum of anti-divisors is palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 242, 252, 323, 434, 727, 4774, 32223, 42024, 43234, 46864, 64946, 70607, 4855584, 4942494, 6125216, 6265626, 149939941, 188737881, 241383142, 389181983, 470212074, 27685458672, 42685658624, 45625352654, 61039793016

Offset: 1

Jason Earls, Sep 03 2002

See A066272 for definition of anti-divisor.

Sean A. Irvine, Table of n, a(n) for n = 1..45

Cf. A002113 (palindromes), A066272, A066417.

from itertools import chain
def is_palindrome(x):
    return x == x[::-1]
A073956 = sorted([n for n in chain(map(lambda x:int(str(x)+str(x)[::-1]),range(1,10**2)),map(lambda x:int(str(x)+str(x)[-2::-1]), range(1,10**3))) if is_palindrome(str(int(sum([d for d in range(2,n,2) if n%d and not 2*n%d]))+int(sum([d for d in range(3,n,2) if n%d and 2*n%d in [d-1,1]]))))]) # Chai Wah Wu, Aug 09 2014

a(21)-a(33) from Donovan Johnson, Mar 30 2010

A093396 Denominators of n divided by the product of the anti-divisors of n.

Original entry on oeis.org

2, 3, 6, 2, 30, 15, 4, 42, 42, 10, 270, 54, 8, 33, 2310, 280, 78, 78, 8, 4050, 4050, 14, 1428, 102, 440, 6270, 114, 32, 7938, 257985, 520, 138, 552, 16, 11250, 866250, 616, 1458, 1458, 2720, 14790, 174, 131040, 16926, 17670, 190, 39204, 78408, 8, 2315250

Offset: 3

Lior Manor, Mar 28 2004

See A066272 for definition of anti-divisor.

			The anti-divisors of 18 are 4, 5, 7, 12. Hence a(18) = 4*5*7*12/GCD(4*5*7*12, 18) = 280.

Dumitru Damian, Table of n, a(n) for n = 3..10000

Cf. A066417, A091507, A093394, A093395 (numerators).

import numpy as np
from sympy.ntheory.factor_ import antidivisors
def a093396(k):
        return (m:=np.prod(antidivisors(k), dtype=object))//np.gcd(m,k, dtype=object)
{print(a093396(k), end = ', ') for k in range(3,10**2)} # Dumitru Damian, Oct 16 2023

a(n) = A091507(n)/GCD(n, A091507(n))

Name changed by Franklin T. Adams-Watters, Aug 21 2013

A210732 Numbers n for which sigma(n)=sigma(x)+sigma(y), where n=x+y and sigma(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

6, 9, 15, 18, 21, 24, 27, 30, 31, 33, 37, 39, 43, 44, 46, 47, 53, 56, 57, 62, 65, 66, 70, 73, 74, 75, 76, 78, 81, 83, 86, 88, 90, 91, 92, 93, 97, 99, 102, 103, 106, 107, 109, 110, 114, 116, 117, 118, 119, 121, 122, 123, 125, 126, 127, 129, 131, 133, 135, 136

Offset: 3

Paolo P. Lava, May 10 2012

nonn

Similar to A211223 but using anti-divisors.

			sigma*(127)=sigma*(45)+sigma*(82) that is 212=86+126.
In more than one way:
sigma*(133)=sigma*(50)+sigma*(83)=sigma*(52)+sigma*(81) that is
204=80+124=94+110.

Cf. A066272, A066417, A083207, A204830, A204831, A211223-A211225.

with(numtheory);
A210732:=proc(q)
local a,b,c,i,j,k,n;
for n from 3 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;
  for i from 1 to trunc(n/2) do
   b:=0; c:=0;
   for k from 2 to i-1 do if abs((i mod k)-k/2)<1 then b:=b+k; fi; od;
   for k from 2 to n-i-1 do if abs(((n-i) mod k)-k/2)<1 then c:=c+k; fi; od;
   if a=b+c then print(n); break; fi;
  od;
od; end:
A210732(10000);

A240979 Sum of unitary anti-divisors of n.

Original entry on oeis.org

0, 0, 2, 3, 5, 0, 10, 8, 2, 10, 12, 5, 19, 12, 2, 14, 28, 12, 18, 16, 2, 32, 34, 7, 29, 20, 18, 38, 24, 0, 42, 58, 20, 26, 28, 0, 50, 66, 20, 39, 41, 22, 56, 32, 22, 54, 60, 24, 58, 56, 2, 86, 88, 0, 42, 40, 30, 92, 90, 35, 57, 74, 32, 46, 48, 26, 132, 104, 2

Offset: 1

Paolo P. Lava, Aug 06 2014

nonn

For unitary anti-divisors of n are intended all the anti-divisors of n which are coprime to n.

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

Cf. A066272, A066417, A242029.

P:=proc(q) local a,k,n,v; v:=[]; 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 if gcd(n,k)=1
then a:=a+k; fi; fi; od; v:=[op(v),a]; od; print(op(v)); end: P(69);
# corrected by Paolo P. Lava, Aug 17 2024

antiDivisors[n_Integer] := Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; a240979[n_Integer] := Total[Select[antiDivisors[n], CoprimeQ[#, n] &]]; a240979 /@ Range[120] (* _Michael De Vlieger, Aug 17 2014 *)

Anti-divisors of 14 are 3, 4, 9. Anti-divisors coprime to 14 are 3 and 9 and therefore a(14) = 3 + 9 = 12.

cp's OEIS Frontend

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

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

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A091507 Product of the anti-divisors of n.

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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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A066418 Numbers k for which phi(k) + anti-phi(k) = k.

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A073956 Palindromes whose sum of anti-divisors is palindromic.

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A093396 Denominators of n divided by the product of the anti-divisors of n.

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.

A210732 Numbers n for which sigma(n)=sigma(x)+sigma(y), where n=x+y and sigma(n) is the sum of the anti-divisors of n.