

A046528


Numbers that are a product of distinct Mersenne primes (elements of A000668).


38



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
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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 Problèmes en Théorie Classique des Nombres, Problem 264 pp. 188, Ellipses Paris 2004.
R. Sivaramakrishnan, Classical Theory of Arithmetic Functions. Dekker, 1989.


LINKS

C. D. H. Cooper, Problem E 2493, The American Mathematical Monthly, Vol. 81, No. 8 (1974), p. 902; W. J. Dodge, solution, ibid., Vol. 82, No. 8 (1975), pp. 855856.


FORMULA



EXAMPLE

a(20) = 82677 = 3*7*31*127, whose sum of divisors is 131072 = 2^17;
a(27) = 1040257 = 127*8191, whose sum of divisors is 1048576 = 2^20.


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



KEYWORD

nonn


AUTHOR



EXTENSIONS

Further terms from Jon Hart, Sep 22 2006


STATUS

approved



