cp's OEIS Frontend

A247825 Numbers which are the difference between the sum of their bi-unitary divisors and the sum of their unitary divisors.

%I A247825 #23 Oct 02 2018 08:18:08
%S A247825 24,240,360
%N A247825 Numbers which are the difference between the sum of their bi-unitary divisors and the sum of their unitary divisors.
%C A247825 No further terms up to 10^8. Is there a relation with 6, 60 and 90, the 3 only bi-unitary perfects? - _Michel Marcus_, Oct 05 2014
%C A247825 a(4), if it exists, is larger than 10^11. - _Giovanni Resta_, Apr 15 2017
%H A247825 C. R. Wall, <a href="http://dx.doi.org/10.1090/S0002-9939-1972-0289403-9">Bi-unitary perfect numbers</a>, Proc. Am. Math. Soc. 33 (1) (1972) 39-42.
%H A247825 Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>
%e A247825 Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Unitary divisors are 1, 3, 8, 24 and their sum is 36. Bi-unitary divisors are 1, 2, 3, 4, 6, 8, 12, 24 and their sum is 60. Then 60 - 36 = 24.
%e A247825 Divisors of 240 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240. Unitary divisors are 1, 3, 5, 15, 16, 48, 80, 240 and their sum is 408. Bi-unitary divisors are 1, 2, 3, 5, 6, 8, 10, 15, 16, 24, 30, 40, 48, 80, 120, 240 and their sum is 648. Then 648 - 408 = 240.
%p A247825 Q:=proc(n) local a, e, p, f; a:=1 ;for f in ifactors(n)[2] do e:=op(2,f); p:=op(1,f);
%p A247825 if type(e,odd) then a:=a*(p^(e+1)-1)/(p-1); else a:=a*((p^(e+1)-1)/(p-1)-p^(e/2)); fi; od: a ; end:
%p A247825 P:=proc(h) local a,b,k,n;
%p A247825 for n from 1 to h do a:=divisors(n); b:=0;
%p A247825 for k from 1 to nops(a) do if gcd(a[k],n/a[k])=1 then b:=b+a[k]; fi; od;
%p A247825 if Q(n)-b=n then print(n); fi; od; end: P(10^6);
%o A247825 (PARI) up(p, e) = p^e+1;
%o A247825 bup(p, e) = my(ret = (p^(e+1) - 1)/(p-1)); if ((e % 2) == 0, ret -= p^(e/2)); ret;
%o A247825 isok(n) = f = factor(n); n == (prod(k=1, #f~, bup(f[k,1], f[k,2])) - prod(k=1, #f~, up(f[k,1], f[k,2]))); \\ _Michel Marcus_, Oct 05 2014
%Y A247825 Cf. A034448, A064591, A188999.
%K A247825 nonn,more,bref
%O A247825 1,1
%A A247825 _Paolo P. Lava_, Sep 29 2014