A192290 Anti-amicable numbers.

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

Offset: 1

Author

Paolo P. Lava, Jun 29 2011

Keywords

• nonn
• more

Comments

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

Examples

			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.
		

Crossrefs

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

Programs

Maple

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);

Python

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

Extensions

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