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 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 (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 Problèmes en Théorie Classique des Nombres, Problem 264 pp. 188, Ellipses Paris 2004. R. Sivaramakrishnan, Classical Theory of Arithmetic Functions. Dekker, 1989. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from T. D. Noe) Kevin S. Brown, Sum of Divisors Equals a Power of 2. 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. 855-856. Jeffrey Shallit, Problem 1319, Diophantine Equation, sigma(n) = 2^m, Math. Magazine, 63 (1990), 129. Eric Weisstein's World of Mathematics, Divisor Function. FORMULA Sum_{n>=1} 1/a(n) = Product_{p in A000668} (1 + 1/p) = 1.5855588879... (A306204) - Amiram Eldar, Jan 06 2021 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())); # 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), A306204. Sequence in context: A108102 A357898 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 November 30 18:31 EST 2023. Contains 367461 sequences. (Running on oeis4.)