login
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.
3

%I #12 Sep 28 2015 03:31:54

%S 10,14,32,58,154,182,382,758,3830,5962,67815454,94941602,7172169026,

%T 8196764584,18624907238,34790550682,30033199624,31387575416,

%U 38857270202,48571587730

%N 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.

%C a(21) > 10^11. - _Hiroaki Yamanouchi_, Sep 28 2015

%e sigma*(10) = 3+4+7 = 14.

%e sigma(14)-14 = 1+2+7 = 10.

%e sigma*(32)= 3+5+7+9+13+21 = 58.

%e sigma(58)-58 = 1+2+29 = 32.

%p with(numtheory);

%p P:= proc(n)

%p local a,b,c,i,ks;

%p for i from 3 to n do

%p a:={};

%p for k from 2 to i-1 do

%p if abs((i mod k)- k/2) < 1 then

%p a:=a union {k};

%p fi;

%p od;

%p b:=nops(a); c:=op(a); s:=0;

%p for k from 1 to b do

%p s:=s+c[k];

%p od;

%p if sigma(s)-s=i then

%p print(i,s);

%p fi;

%p od;

%p end:

%p P(10000);

%Y Cf. A063990, A066272, A192290, A192292, A192293.

%K nonn,more

%O 1,1

%A _Paolo P. Lava_, Jun 29 2011

%E a(11)-a(20) from _Hiroaki Yamanouchi_, Sep 28 2015