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

Author

Labos Elemer, May 22 2000

Keywords

• nonn

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)

Examples

			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
		

Links

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

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.

Programs

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 *)

PARI

A209229(n) = (n && !bitand(n,n-1));
isA054784(n) = A209229(sigma(n>>valuation(n,2))); \\ Antti Karttunen, Aug 28 2021

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