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
A192291
Couple of numbers a, b for which sigma*(a)=b and sigma(b)-b=a, where sigma*(n) is the sum of the anti-divisors of n.
Original entry on oeis.org
10, 14, 32, 58, 154, 182, 382, 758, 3830, 5962, 67815454, 94941602, 7172169026, 8196764584, 18624907238, 34790550682, 30033199624, 31387575416, 38857270202, 48571587730
Offset: 1
sigma*(10) = 3+4+7 = 14.
sigma(14)-14 = 1+2+7 = 10.
sigma*(32)= 3+5+7+9+13+21 = 58.
sigma(58)-58 = 1+2+29 = 32.
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with(numtheory);
P:= proc(n)
local a,b,c,i,ks;
for i from 3 to n do
a:={};
for k from 2 to i-1 do
if abs((i mod k)- k/2) < 1 then
a:=a union {k};
fi;
od;
b:=nops(a); c:=op(a); s:=0;
for k from 1 to b do
s:=s+c[k];
od;
if sigma(s)-s=i then
print(i,s);
fi;
od;
end:
P(10000);
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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