cp's OEIS Frontend

A273813 Composite numbers whose sum of unitary divisors is a multiple of the sum of their aliquot parts.

%I A273813 #22 Jan 14 2019 06:11:39
%S A273813 6,24,112,1984,32512,171197,667879,780625,56513539,134201344,
%T A273813 488265625,5203009849,9130639447,34359476224,47390685029,96595448129
%N A273813 Composite numbers whose sum of unitary divisors is a multiple of the sum of their aliquot parts.
%C A273813 A064591 is a subsequence of this sequence.
%C A273813 The ratios are 2, 1, 1, 1, 1, 12, 16, 4, 40, 1, 4, 100, 112, 1, 156, 180, ...
%C A273813 Up to 3*10^11 all the terms are of the form p^e*q. In particular, if 2^k-1 is prime, then 2^(k+1)(2^k-1) is a term. Similarly, if 2*5^k-1 is prime, then 5^k*(2*5^k-1) is a term. By solving appropriate Diophantine equations it is also possible to obtain large terms of the form p^2*q, like 1300253^2*1099140634496715133. - _Giovanni Resta_, Jun 01 2016
%e A273813 Unitary divisors of 6 are 1, 2, 3, 6 and their sum is 12. Aliquot parts are 1, 2, 3 and their sum is 6.  Then, 12 / 6 = 2.
%e A273813 Unitary divisors of 24 are 1, 3, 8, 24 and their sum is 36. Aliquot parts are 1, 2, 3, 4, 6, 8, 12 and their sum is 36.  Then, 36 / 36 = 1.
%e A273813 Unitary divisors of 171197 are 1, 169, 1013, 171197 and their sum is 172380. Aliquot parts are 1, 13, 169, 1013, 13169 and their sum is 14365.  Then, 172380 / 14365 = 12.
%p A273813 with(numtheory): P:=proc(q) local a,b,k,n;
%p A273813 for n from 2 to q do if not isprime(n) then a:=ifactors(n)[2]; b:=mul(a[k][1]^a[k][2]+1,k=1..nops(a));
%p A273813 if type(b/(sigma(n)-n),integer) then print(n); fi; fi; od; end: P(10^9);
%t A273813 Select[Range[10^6], Function[n, CompositeQ@ n && Mod[Total@ Select[Divisors@ n, GCD[#, n/#] == 1 &], DivisorSigma[1, n] - n] == 0]] (* _Michael De Vlieger_, Jun 01 2016 *)
%Y A273813 Cf. A001065, A034448, A064591.
%K A273813 nonn,more
%O A273813 1,1
%A A273813 _Paolo P. Lava_, May 31 2016
%E A273813 a(9)-a(16) from _Giovanni Resta_, Jun 01 2016