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

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

Links

Table of n, a(n) for n=1..16.

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

Cf. A001065, A034448, A064591.

Sequence in context: A038380 A052745 A187668 * A293257 A223752 A293236

Adjacent sequences: A273810 A273811 A273812 * A273814 A273815 A273816

Keyword

nonn,more

Author

Paolo P. Lava, May 31 2016

Extensions

a(9)-a(16) from Giovanni Resta, Jun 01 2016

Status

approved