

A054784


Integers n such that sigma(2n)  sigma(n) is a power of 2, where sigma is the sum of the divisors of n.


2



1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 31, 32, 42, 48, 56, 62, 64, 84, 93, 96, 112, 124, 127, 128, 168, 186, 192, 217, 224, 248, 254, 256, 336, 372, 381, 384, 434, 448, 496, 508, 512, 651, 672, 744, 762, 768, 868, 889, 896, 992, 1016, 1024, 1302, 1344, 1488
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OFFSET

1,2


COMMENTS

If n is a squarefree product of Mersenne primes multiplied by a power of 2, then sigma(2n)  sigma(n) is a power of 2.
The reverse is also true. All numbers in this sequence have this form.  Ivan Neretin, Aug 12 2016


LINKS

Ivan Neretin, Table of n, a(n) for n = 1..10000


FORMULA

Numbers n such that A000203(2*n)  A000203(n) = 2^w for some w.


EXAMPLE

For n=12, sigma(2n) = sigma(24) = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 and sigma(n) = sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. So sigma(2n)  sigma(n) = 60  28 = 32 = 2^5 is a power of 2, and therefore 12 is in the sequence.  Michael B. Porter, Aug 15 2016


MAPLE

N:= 10^6: # to get all terms <= N
M:= select(isprime, [seq(2^i1, i=select(isprime, [$2..ilog2(N+1)]))]):
R:= map(t > seq(2^i*t, i=0..floor(log[2](N/t))), map(convert, combinat:powerset(M), `*`)):
sort(convert(R, list)); # Robert Israel, Aug 12 2016


MATHEMATICA

Sort@Select[Flatten@Outer[Times, p2 = 2^Range[0, 11], Times @@ # & /@ Subsets@Select[p2  1, PrimeQ]], # <= Max@p2 &] (* Ivan Neretin, Aug 12 2016 *)
Select[Range[1500], IntegerQ[Log2[DivisorSigma[1, 2#]DivisorSigma[1, #]]]&] (* Harvey P. Dale, Apr 23 2019 *)


CROSSREFS

Cf. A000203, A000668.
Sequence in context: A277704 A082752 A023758 * A018585 A018399 A329855
Adjacent sequences: A054781 A054782 A054783 * A054785 A054786 A054787


KEYWORD

nonn


AUTHOR

Labos Elemer, May 22 2000


STATUS

approved



