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A046528 Numbers that are a product of distinct Mersenne primes (elements of A000668). 20
1, 3, 7, 21, 31, 93, 127, 217, 381, 651, 889, 2667, 3937, 8191, 11811, 24573, 27559, 57337, 82677, 131071, 172011, 253921, 393213, 524287, 761763, 917497, 1040257, 1572861, 1777447, 2752491, 3120771, 3670009, 4063201, 5332341, 7281799, 11010027, 12189603 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Or, numbers n such that the sum of the divisors of n is a power of 2, see A094502.

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>1 such that sigma(n)=2*phi(sigma(n)) - Benoit Cloitre, Feb 22 2002

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. - T. D. Noe, Oct 12 2006

Supersequence of A051281 (numbers n such that sigma(n) is a power of tau(n)). Conjecture: numbers n such that sigma(n) = tau(n)^(a/b), where a, b are integers >= 1. Example: sigma(93) = 128 = tau(93)^(7/2) = 4^(7/2). - Jaroslav Krizek, May 04 2013

REFERENCES

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

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

LINKS

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

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

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

Eric Weisstein's World of Mathematics, Divisor Function.

EXAMPLE

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

MAPLE

mersennes:= [seq(numtheory:-mersenne([i]), i=1..10)]:

sort(select(`<`, map(convert, combinat:-powerset(mersennes), `*`), numtheory:-mersenne([11]))); # Robert Israel, May 01 2016

MATHEMATICA

{1}~Join~TakeWhile[Times @@@ Rest@ Subsets@ # // Sort, Function[k, k <= Last@ #]] &@ Select[2^Range[0, 31] - 1, PrimeQ] (* Michael De Vlieger, May 01 2016 *)

PROG

(PARI) isok(n) = (n==1) || (ispower(sigma(n), , &r) && (r==2)); \\ Michel Marcus, Dec 10 2013

CROSSREFS

Cf. A000668, A000043, A056652 (product of Mersenne primes).

Sequence in context: A003585 A108102 A065523 * A018572 A018641 A097162

Adjacent sequences:  A046525 A046526 A046527 * A046529 A046530 A046531

KEYWORD

nonn

AUTHOR

Labos Elemer

EXTENSIONS

More terms from Benoit Cloitre, Feb 22 2002

Further terms from Jon Hart, Sep 22 2006

Entry revised by N. J. A. Sloane, Mar 20 2007

Three more terms from Michel Marcus, Dec 10 2013

STATUS

approved

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Last modified June 22 12:24 EDT 2017. Contains 288613 sequences.