A066272 -id:A066272 - cp's OEIS frontend

A222565 Primes that are the largest anti-divisor of primes.

2, 3, 5, 7, 11, 13, 19, 29, 31, 41, 47, 53, 59, 67, 71, 73, 101, 109, 127, 131, 149, 151, 167, 179, 181, 211, 233, 239, 281, 293, 307, 311, 347, 349, 379, 401, 409, 421, 431, 439, 449, 461, 467, 479, 541, 547, 569, 571, 587, 607, 613, 619, 631, 647, 661, 673, 701

Offset: 1

Juri-Stepan Gerasimov, Feb 25 2013

nonn

See A066272 for definition of anti-divisor.

Primes p such that 2p + largest anti-divisor of 2p is also prime: 2, 5, 7, 11, 13, 29, 31, 41, 47, 59, 67, 79, 83, 101, 137, 139, 151, 157, 167, 173, 193, 223, 227, 239, 257,...

			The prime 19 is here because it is largest anti-divisor of prime 29.

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

Cf. A066481.

is(n)=isprime(n)&&isprime(bitor((3*n-1)\2,1)) \\ Charles R Greathouse IV, Feb 27 2013

2 together with primes of the form 4k+1 such that 6k+1 is prime, together with primes of the form 4k+3 such that 6k+5 is prime. - Charles R Greathouse IV, Feb 27 2013

Missing terms a(9), a(21), a(28), a(29) added by Charles R Greathouse IV, Feb 27 2013

A066481 Largest anti-divisor of n.

Original entry on oeis.org

2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 47, 47, 48, 49, 49, 50

Offset: 3

Robert G. Wilson v, Jan 02 2002

nonn

Apart from initial terms this is identical to A004396.

See A066272 for definition of anti-divisor.

Seiichi Manyama, Table of n, a(n) for n = 3..10000

Cf. A066482.

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:
A066481 := proc(n)
if n < 3 then
    return 0;
else
    sort(convert(antidivisors(n),list)) ;
    op(-1,%) ;
end if;
end proc: # R. J. Mathar, Mar 15 2013

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[ Last[ antid[n]], {n, 3, 100} ]

a(n)=2*n\/3 \\ Charles R Greathouse IV, Feb 27 2013

A130846 Replace n with the concatenation of its anti-divisors.

Original entry on oeis.org

2, 3, 23, 4, 235, 35, 26, 347, 237, 58, 2359, 349, 2610, 311, 235711, 45712, 2313, 3813, 2614, 345915, 235915, 716, 2371017, 3417, 2561118, 3581119, 2319, 41220, 237921, 35791321, 2561322, 3423, 23101423, 824, 2351525, 3457111525, 2671126, 391627

Offset: 3

Jonathan Vos Post, Jul 20 2007, Jul 22 2007

Number of anti-divisors concatenated to form a(n) is A066272(n). We may consider prime values of the concatenated anti-divisor sequence and we may iterate it, i.e. n, a(n), a(a(n)), a(a(a(n))) which leads to questions of trajectory, cycles, fixed points.
See A066272 for definition of anti-divisor.
Primes in this sequence are at n=3,4,5,10,14,16,40,46,100,145,149,... - R. J. Mathar, Jul 24 2007

			3: 2, so a(3) = 2.
4: 3, so a(4) = 3.
5: 2, 3, so a(5) = 23.
6: 4, so a(6) = 4.
7: 2, 3, 5, so a(7) = 235.
17: 2, 3, 5, 7, 11, so a(17) = 235711

Jon Perry, The Anti-Divisor, cached copy.
Jonathan Vos Post, Factors of first 62 terms

Cf. A037278, A066272, A120712, A106708, A130799.

antiDivs := proc(n) local resul,odd2n,r ; resul := {} ; for r in ( numtheory[divisors](2*n-1) union numtheory[divisors](2*n+1) ) do if n mod r <> 0 and r> 1 and r < n then resul := resul union {r} ; fi ; od ; odd2n := numtheory[divisors](2*n) ; for r in odd2n do if ( r mod 2 = 1) and r > 2 then resul := resul union {2*n/r} ; fi ; od ; RETURN(resul) ; end: A130846 := proc(n) cat(op(antiDivs(n))) ; end: seq(A130846(n),n=3..80) ; # R. J. Mathar, Jul 24 2007

from sympy.ntheory.factor_ import antidivisors
def A130846(n): return int(''.join(str(s) for s in antidivisors(n))) # Chai Wah Wu, Dec 08 2021

More terms from R. J. Mathar, Jul 24 2007

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

A241557 Numbers k that do not have prime anti-divisors.

Original entry on oeis.org

1, 2, 6, 30, 36, 54, 90, 96, 114, 120, 156, 174, 210, 216, 300, 330, 414, 510, 516, 546, 576, 660, 714, 726, 744, 804, 810, 834, 894, 936, 966, 1014, 1044, 1056, 1134, 1170, 1296, 1344, 1356, 1500, 1560, 1584, 1626, 1650, 1680, 1686, 1734, 1764, 1770, 1836, 1884, 1926, 2010, 2046, 2064

Offset: 1

Michael De Vlieger, Aug 08 2014

nonn

			a(3) = 6, since 6 has the anti-divisor 4, and it is composite.
a(4) = 30, since 30 has the anti-divisors {4, 12, 20} and none are prime.
All the integers 6 < k < 30 have at least one prime anti-divisor, and the only integers k < 6 that do not have prime antidivisors are k = {1, 2}.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n=1...167 from Michael De Vlieger)

Cf. A066272, A241556.

primeAntiDivisors[n_] := Select[Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)], PrimeQ]; a241556[n_Integer] := Map[Length[primeAntiDivisors[#]] &, Range[n]]; Flatten[Position[a241556[10^5],0]]

from sympy import isprime, divisors
A241557 = [n for n in range(1,10**6) if not any([isprime(x) for x in
          [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]])]
# Chai Wah Wu, Aug 19 2014

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

A191580 Numbers n for which the sum of their prime factors (with repetition) divides the sum of their anti-divisors.

Original entry on oeis.org

5, 10, 40, 41, 129, 135, 140, 155, 182, 189, 200, 204, 206, 238, 375, 429, 435, 441, 455, 475, 546, 564, 574, 616, 625, 678, 722, 744, 765, 836, 856, 902, 1035, 1056, 1170, 1188, 1272, 1296, 1344, 1518, 1650, 1764, 1806, 1918, 1925

Offset: 1

Paolo P. Lava, Jun 07 2011

			40-> sum prime factors=2+2+2+5=11; sum anti-divisors=3+9+16+27=55; 55/11=5
129-> sum prime factors=3+43=46; sum anti-divisors=2+6+7+37+86=138; 138/46=3

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

Cf. A001414, A066272, A161917, A191581.

with(numtheory); P:=proc(i) local a,b,j,k,s,n;
for n from 3 to i do b:=ifactors(n)[2];
s:=add(b[k][1]*b[k][2],k=1..nops(b));
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 type(a/s,integer) then print(n); fi; od; end: P(2000);

cp's OEIS Frontend

A222565 Primes that are the largest anti-divisor of primes.

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A066481 Largest anti-divisor of n.

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A130846 Replace n with the concatenation of its anti-divisors.

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A192270 Pseudo anti-perfect numbers.

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A192290 Anti-amicable numbers.

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A241557 Numbers k that do not have prime anti-divisors.

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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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