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A046528 Numbers that are a product of distinct Mersenne primes (elements of A000668).

%I A046528 #77 Feb 16 2025 08:32:39
%S A046528 1,3,7,21,31,93,127,217,381,651,889,2667,3937,8191,11811,24573,27559,
%T A046528 57337,82677,131071,172011,253921,393213,524287,761763,917497,1040257,
%U A046528 1572861,1777447,2752491,3120771,3670009,4063201,5332341,7281799,11010027,12189603
%N A046528 Numbers that are a product of distinct Mersenne primes (elements of A000668).
%C A046528 Or, numbers n such that the sum of the divisors of n is a power of 2, see A094502.
%C A046528 Or, numbers n such that the number of divisors of n and the sum of the divisors of n are both powers of 2.
%C A046528 n is a product of distinct Mersenne primes iff sigma(n) is a power of 2: see exercise in Sivaramakrishnan, or Shallit.
%C A046528 Sequence gives n > 1 such that sigma(n) = 2*phi(sigma(n)). - _Benoit Cloitre_, Feb 22 2002
%C A046528 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
%C A046528 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
%D A046528 J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problem 264 pp. 188, Ellipses Paris 2004.
%D A046528 R. Sivaramakrishnan, Classical Theory of Arithmetic Functions. Dekker, 1989.
%H A046528 Amiram Eldar, <a href="/A046528/b046528.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from T. D. Noe)
%H A046528 Kevin S. Brown, <a href="https://www.mathpages.com/home/kmath048/kmath048.htm">Sum of Divisors Equals a Power of 2</a>.
%H A046528 C. D. H. Cooper, <a href="http://www.jstor.org/stable/2319455">Problem E 2493</a>, The American Mathematical Monthly, Vol. 81, No. 8 (1974), p. 902; W. J. Dodge, <a href="http://www.jstor.org/stable/2319819">solution</a>, ibid., Vol. 82, No. 8 (1975), pp. 855-856.
%H A046528 Jeffrey Shallit, <a href="http://www.jstor.org/stable/2691073">Problem 1319, Diophantine Equation, sigma(n) = 2^m</a>, Math. Magazine, 63 (1990), 129.
%H A046528 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>.
%F A046528 Sum_{n>=1} 1/a(n) =  Product_{p in A000668} (1 + 1/p) = 1.5855588879... (A306204) - _Amiram Eldar_, Jan 06 2021
%e A046528 a(20) = 82677 = 3*7*31*127, whose sum of divisors is 131072 = 2^17;
%e A046528 a(27) = 1040257 = 127*8191, whose sum of divisors is 1048576 = 2^20.
%p A046528 mersennes:= [seq(numtheory:-mersenne([i]),i=1..10)]:
%p A046528 sort(select(`<`,map(convert,combinat:-powerset(mersennes),`*`),numtheory:-mersenne([11]))); # _Robert Israel_, May 01 2016
%t A046528 {1}~Join~TakeWhile[Times @@@ Rest@ Subsets@ # // Sort, Function[k, k <= Last@ #]] &@ Select[2^Range[0, 31] - 1, PrimeQ] (* _Michael De Vlieger_, May 01 2016 *)
%o A046528 (PARI) isok(n) = (n==1) || (ispower(sigma(n), , &r) && (r==2)); \\ _Michel Marcus_, Dec 10 2013
%Y A046528 Cf. A000668, A000043, A056652 (product of Mersenne primes), A306204.
%K A046528 nonn
%O A046528 1,2
%A A046528 _Labos Elemer_
%E A046528 More terms from _Benoit Cloitre_, Feb 22 2002
%E A046528 Further terms from _Jon Hart_, Sep 22 2006
%E A046528 Entry revised by _N. J. A. Sloane_, Mar 20 2007
%E A046528 Three more terms from _Michel Marcus_, Dec 10 2013