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A273813
Composite numbers whose sum of unitary divisors is a multiple of the sum of their aliquot parts.
0
6, 24, 112, 1984, 32512, 171197, 667879, 780625, 56513539, 134201344, 488265625, 5203009849, 9130639447, 34359476224, 47390685029, 96595448129
OFFSET
1,1
COMMENTS
A064591 is a subsequence of this sequence.
The ratios are 2, 1, 1, 1, 1, 12, 16, 4, 40, 1, 4, 100, 112, 1, 156, 180, ...
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
EXAMPLE
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.
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.
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.
MAPLE
with(numtheory): P:=proc(q) local a, b, k, n;
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));
if type(b/(sigma(n)-n), integer) then print(n); fi; fi; od; end: P(10^9);
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Paolo P. Lava, May 31 2016
EXTENSIONS
a(9)-a(16) from Giovanni Resta, Jun 01 2016
STATUS
approved