1,2

Or, numbers n such that the sum of the divisors of n is a power of 2.

Or, numbers n such that the number of divisors of n and the sum of the divisors of n are both powers of 2.

n is a product of distinct Mersenne primes iff sigma(n) is a power of 2: see exercise in Sivaramakrishnan, or Shallit.

Sequence gives n such that sigma(n)=2*phi(sigma(n)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 22 2002

Comment from T. D. Noe, Oct 12 2006: The graph of this sequence shows a discontinuity at the 4097th number because there is a large relative gap between the 12th and 13th Mersenne primes, A000043. Other discontinuities can be predicted using A078426.

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 264 pp. 188, Ellipses Paris 2004.

J. Shallit, Problem 1319, Diophantine Equation, sigma(n) = 2^m, Math. Magazine, 63 (1990), 129.

R. Sivaramakrishnan, Classical Theory of Arithmetic Functions. Dekker, 1989.

T. D. Noe, Table of n, a(n) for n=1..5000

K. S. Brown, Sum of Divisors Equals a Power of 2

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

a[ 27 ]=1040257=7*127*8191*131071 and Sum[ d ]=1048576 a[ 20 ]=82677=3*7*31*127 and Sum[ d ]=131072

Cf. A000668, A000043, A046528.

Sequence in context: A003585 A108102 A065523 this_sequence A018572 A018641 A097162

Adjacent sequences: A046525 A046526 A046527 this_sequence A046529 A046530 A046531

nonn

Labos E. (labos(AT)ana.sote.hu)

More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 22 2002

Further terms from Jon Hart, Sep 22 2006

Entry revised by njas, Mar 20 2007