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
From Antti Karttunen, Sep 01 2021: (Start)
Numbers k such that the sum of their odd divisors [A000593(k)] is a power of 2.
(End)
LINKS
Ivan Neretin, Table of n, a(n) for n = 1..10000
FORMULA
Sum_{n>=1} 1/a(n) = 2 * Product_{p in A000668} (1 + 1/p) = 2 * A306204 = 3.1711177758... . - Amiram Eldar, Jan 11 2023
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^i-1, 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 *)
PROG
(PARI)
A209229(n) = (n && !bitand(n, n-1));
isA054784(n) = A209229(sigma(n>>valuation(n, 2))); \\ Antti Karttunen, Aug 28 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, May 22 2000
STATUS
approved