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);
-
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
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);
A229860
Let sigma*_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma*_m (n) = k*n; sequence gives the (2,k)-anti-perfect numbers.
Original entry on oeis.org
3, 5, 7, 8, 14, 16, 32, 41, 56, 92, 98, 114, 167, 507, 543, 946, 2524, 3433, 5186, 5566, 6596, 6707, 6874, 8104, 9615, 15690, 17386, 27024, 84026, 87667, 167786, 199282, 390982, 1023971, 1077378, 1336968, 1529394, 2054435, 2276640, 2667584, 3098834, 3978336
Offset: 1
Anti-divisors of 92 are 3, 5, 8, 37, 61. Their sum is 114.
Again, anti-divisors of 114 are 4, 12, 76. Their sum is 92 and 92 / 92 = 1.
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with(numtheory); P:=proc(q,h) local a,i,j,k,n;
for n from 3 to q do a:=n; for i from 1 to h do
k:=0; j:=a; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*a+1)+sigma(2*a-1)+sigma(a/2^k)*2^(k+1)-6*a-2; od;
if type(a/n,integer) then print(n); fi; od; end: P(10^6,2);
A229861
Let sigma*_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma*_m (n) = k*n; sequence gives the (3,k)-anti-perfect numbers.
Original entry on oeis.org
4, 5, 8, 32, 41, 54, 56, 68, 123, 946, 1494, 1856, 2056, 5186, 6874, 8104, 10419, 17386, 27024, 31100, 84026, 167786, 272089, 733253, 812600, 1188000, 1544579, 2667584, 4921776, 16360708, 21524990, 27914146
Offset: 1
Anti-divisors of 54 are 4, 12, 36. Their sum is 52.
Again, anti-divisors of 52 are 3, 5, 7, 8, 15, 21, 35. Their sum is 94.
Finally, anti-divisors of 94 are 3, 4, 7, 9, 11, 17, 21, 27, 63. Their sum is 162 and 162 / 54 = 3.
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with(numtheory); P:=proc(q,h) local a,i,j,k,n;
for n from 4 to q do a:=n; for i from 1 to h do
k:=0; j:=a; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*a+1)+sigma(2*a-1)+sigma(a/2^k)*2^(k+1)-6*a-2; od;
if type(a/n,integer) then print(n); fi; od; end: P(10^6,3);
A229862
Let sigma*_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma*_m (n) = k*n; sequence gives the (4,k)-anti-perfect numbers.
Original entry on oeis.org
5, 6, 7, 8, 14, 16, 41, 46, 56, 58, 64, 92, 96, 114, 946, 3307, 3325, 5186, 5566, 6596, 6874, 7982, 8104, 14621, 17386, 27024, 44217, 45970, 84026, 91282, 135592, 167786, 1077378, 1231058, 1529394, 2667584, 2873910, 3098834, 3978336, 4292594, 4921776, 27914146
Offset: 1
Anti-divisors of 58 are 3, 4, 5, 9, 13, 23, 39. Their sum is 96.
The only anti-divisor of 96 is 64.
Again, anti-divisors of 64 are 3, 43. Their sum is 46. Finally, anti-divisors of 46 are 3, 4, 7, 13, 31. Their sum is 58 and 58 / 58 = 1.
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with(numtheory); P:=proc(q,h) local a,i,j,k,n;
for n from 5 to q do a:=n; for i from 1 to h do
k:=0; j:=a; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*a+1)+sigma(2*a-1)+sigma(a/2^k)*2^(k+1)-6*a-2; od;
if type(a/n,integer) then print(n); fi; od; end: P(10^6,4);
Showing 1-6 of 6 results.
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