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A054784
Integers n such that sigma(2n) - sigma(n) is a power of 2, where sigma is the sum of the divisors of n.
11
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
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.
Numbers k whose odd part [A000265(k)] is in A046528.
(End)
LINKS
FORMULA
Numbers n such that A000203(2*n) - A000203(n) = 2^w for some w.
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
Cf. A000203, A000265, A000396 (even terms form a subsequence), A000593, A000668, A046528, A063883, A209229, A306204, A331410, A336923 (characteristic function).
Positions of zeros in A336922. Positions of 0's and 1's in A336361.
Cf. also A003401.
Sequence in context: A277704 A082752 A023758 * A018585 A018399 A329855
KEYWORD
nonn
AUTHOR
Labos Elemer, May 22 2000
STATUS
approved