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
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.
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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);
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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
A192290
Anti-amicable numbers.
Original entry on oeis.org
14, 16, 92, 114, 5566, 6596, 1077378, 1529394, 3098834, 3978336, 70774930, 92974314
Offset: 1
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.
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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);
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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
A192292
Pairs of numbers a, b for which sigma*(a)=b and sigma(b)-b-1=a, where sigma*(n) is the sum of the anti-divisors of n.
Original entry on oeis.org
7, 10, 14, 16, 45, 86, 2379, 2324, 4213, 5866, 27323, 33604, 1303227, 1737628, 3722831, 4208308, 15752651, 18706108, 6094085371, 8114352508, 30090695519, 40119052564
Offset: 1
sigma*(45)= 2+6+7+10+13+18+30 = 86.
sigma(86)-86-1 = 2+43 = 45.
sigma*(2379) = 2+6+26+67+71+78+122+366+1586 = 2374.
sigma(2324)-2324-1 = 2+4+7+14+28+83+166+332+581+1162 = 2379.
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with(numtheory); P:= proc(n) local b,c,i,j,k;
for i from 3 to n do k:=0; j:=i;
while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
b:=sigma(2*i+1)+sigma(2*i-1)+sigma(i/2^k)*2^(k+1)-6*i-2;
if sigma(b)-b-1=i then print(i); print(b); fi;
od; end: P(10^9);
Showing 1-3 of 3 results.
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