login
Numbers k such that sigma*(k) = Sum_{j=anti-divisors of k} sigma*(j), where sigma*(k) is the sum of the anti-divisors of k.
1

%I #17 Feb 14 2021 15:25:37

%S 1,2,11,12,15,16,22,31,76,152,309,1576,375479,781314,1114986,3734218,

%T 24311881,68133239,147881549

%N Numbers k such that sigma*(k) = Sum_{j=anti-divisors of k} sigma*(j), where sigma*(k) is the sum of the anti-divisors of k.

%C Tested up to k = 108122.

%C a(20) > 3*10^8. - _Donovan Johnson_, Mar 22 2013

%e Anti-divisors of 76 are 3, 8, 9, 17 and 51 and their sum is 88.

%e Anti-divisor of 3 is 2 -> Sum is 2.

%e Anti-divisors of 8 are 3 and 5 -> Sum is 8.

%e Anti-divisors of 9 are 2 and 6 -> Sum is 8.

%e Anti-divisors of 17 are 2, 3, 5, 7 and 11 -> Sum is 28.

%e Anti-divisors of 51 are 2, 6 and 34 -> Sum is 42.

%e Finally, 2+8+8+28+42=88.

%p A216213:= proc(q) local a,b,c,j,k,n;

%p for n from 1 to q do

%p a:={}; b:=0; for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then b:=b+k; a:=a union {k}; fi; od;

%p c:=0; for j from 1 to nops(a) do for k from 2 to a[j]-1 do if abs((a[j] mod k)-k/2)<1 then c:=c+k; fi; od; od; if b=c then print(n); fi; od; end:

%p A216213(10^10);

%Y Cf. A066272, A178029, A191580, A191581, A192282-A192284.

%K nonn

%O 1,2

%A _Paolo P. Lava_, Mar 13 2013

%E a(13)-a(19) from _Donovan Johnson_, Mar 22 2013