A054784 Integers n such that sigma(2n) - sigma(n) is a power of 2, where sigma is the sum of the divisors of n.

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

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.

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

Adjacent sequences: A054781 A054782 A054783 * A054785 A054786 A054787

Keyword

nonn

Author

Labos Elemer, May 22 2000

Status

approved