A306204 -id:A306204 - cp's OEIS frontend

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

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

Offset: 1

Labos Elemer

nonn

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

			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.

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.

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.

Cf. A000668, A000043, A056652 (product of Mersenne primes), A306204.

mersennes:= [seq(numtheory:-mersenne([i]),i=1..10)]:
sort(select(`<`,map(convert,combinat:-powerset(mersennes),`*`),numtheory:-mersenne([11]))); # Robert Israel, May 01 2016

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

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

Sum_{n>=1} 1/a(n) = Product_{p in A000668} (1 + 1/p) = 1.5855588879... (A306204) - Amiram Eldar, Jan 06 2021

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

A054784 Integers n such that sigma(2n) - sigma(n) is a power of 2, where sigma is the sum of the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 31, 32, 42, 48, 56, 62, 64, 84, 93, 96, 112, 124, 127, 128, 168, 186, 192, 217, 224, 248, 254, 256, 336, 372, 381, 384, 434, 448, 496, 508, 512, 651, 672, 744, 762, 768, 868, 889, 896, 992, 1016, 1024, 1302, 1344, 1488

Offset: 1

Labos Elemer, May 22 2000

nonn

If n is a squarefree product of Mersenne primes multiplied by a power of 2, then sigma(2n) - sigma(n) is a power of 2.
The reverse is also true. All numbers in this sequence have this form. - Ivan Neretin, Aug 12 2016
From Antti Karttunen, Sep 01 2021: (Start)
Numbers k such that the sum of their odd divisors [A000593(k)] is a power of 2.
Numbers k whose odd part [A000265(k)] is in A046528.
(End)

			For n=12, sigma(2n) = sigma(24) = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 and sigma(n) = sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. So sigma(2n) - sigma(n) = 60 - 28 = 32 = 2^5 is a power of 2, and therefore 12 is in the sequence. - _Michael B. Porter_, Aug 15 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A000265, A000396 (even terms form a subsequence), A000593, A000668, A046528, A063883, A209229, A306204, A331410, A336923 (characteristic function).
Positions of zeros in A336922. Positions of 0's and 1's in A336361.
Cf. also A003401.

N:= 10^6: # to get all terms <= N
M:= select(isprime, [seq(2^i-1, i=select(isprime, [$2..ilog2(N+1)]))]):
R:= map(t -> seq(2^i*t, i=0..floor(log[2](N/t))), map(convert,combinat:-powerset(M),`*`)):
sort(convert(R,list)); # Robert Israel, Aug 12 2016

Sort@Select[Flatten@Outer[Times, p2 = 2^Range[0, 11], Times @@ # & /@ Subsets@Select[p2 - 1, PrimeQ]], # <= Max@p2 &] (* Ivan Neretin, Aug 12 2016 *)
Select[Range[1500],IntegerQ[Log2[DivisorSigma[1,2#]-DivisorSigma[1,#]]]&] (* Harvey P. Dale, Apr 23 2019 *)

A209229(n) = (n && !bitand(n,n-1));
isA054784(n) = A209229(sigma(n>>valuation(n,2))); \\ Antti Karttunen, Aug 28 2021

Numbers n such that A000203(2*n) - A000203(n) = 2^w for some w.

Sum_{n>=1} 1/a(n) = 2 * Product_{p in A000668} (1 + 1/p) = 2 * A306204 = 3.1711177758... . - Amiram Eldar, Jan 11 2023

cp's OEIS Frontend

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

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