cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Labos Elemer

Keywords

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

Examples

			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.

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

Crossrefs

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

Programs

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

Formula

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

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

A094505 Powers of 2 which are not the sum of divisors of any other number. Powers of 2 present in A007369.

Original entry on oeis.org

2, 16, 64, 2048
Offset: 1

Views

Author

Labos Elemer, Jun 02 2004

Keywords

Comments

The next term, 2^470, is too large to include.

Crossrefs

Cf. A007369, A000203, A046528, A094502.

Formula

a(n) = 2^A078426(n).

A247473 Numbers of the form 2^k (k>=0) that are a sum of divisors of n for some n.

Original entry on oeis.org

1, 4, 8, 32, 128, 256, 512, 1024, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368
Offset: 1

Views

Author

Jaroslav Krizek, Feb 27 2015

Keywords

Comments

Set of values A094502(n) = sigma(A046528(n)) in increasing order.
Complement of A094505 with respect to A000079 (powers of 2).
Corresponding values of numbers k>0 are in A180221.

Examples

			32 = 2^5 is in sequence because there are numbers n = 21 and 31 with sigma(n) = 32.

Crossrefs

Cf. A000079, A000203, A046528, A078426, A094502, A094505, A180221.

Programs

  • Magma
    Set(Sort([SumOfDivisors(n): n in [A046528(n)]]))
  • Magma
    Set(Sort([SumOfDivisors(n): n in[1..10000], k in [0..100] | SumOfDivisors(n) eq 2^k]))
  • Magma
    [1] cat [2^n: n in[A180221(n)]]

Formula

a(1) = 1, for n>=2, a(n) = 2^A180221(n-1).
Showing 1-3 of 3 results.

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