A061645 a(n) is the number of divisors of n-th even perfect number.

4, 6, 10, 14, 26, 34, 38, 62, 122, 178, 214, 254, 1042, 1214, 2558, 4406, 4562, 6434, 8506, 8846, 19378, 19882, 22426, 39874, 43402, 46418, 88994, 172486, 221006, 264098, 432182, 1513678, 1718866, 2515574, 2796538, 5952442, 6042754, 13945186, 26933834

Offset: 1

Author

Labos Elemer, Jun 14 2001

Keywords

• nonn

Comments

The number of divisors of n-th perfect number that are powers of 2 is equal to a(n)/2, assuming there are no odd perfect numbers. The number of divisors of n-th perfect number that are multiples of n-th Mersenne prime A000668(n) is also equal to a(n)/2, assuming there are no odd perfect numbers. (See A000043). - Omar E. Pol, Feb 28 2008

The n-th even perfect number A000396(n) = 2^(p-1)*P with Mersenne prime P = 2^p-1, p = A000043(n), has obviously the 2p divisors { 1, 2, 2^2, ..., 2^(p-1) } U { P, 2*P, ..., 2^(p-1)*P }. - M. F. Hasler, Dec 10 2018

Examples

			8128 = 2*2*2*2*2*2*127 with 14 divisors.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)
• Omar E. Pol, Los números perfectos, (in Spanish).

Crossrefs

Cf. A000005, A000043, A000396, A000668.

Programs

Mathematica

2 * Array[MersennePrimeExponent, 45] (* Amiram Eldar, Dec 10 2018 *)

PARI

A061645(n)=2*A000043(n) \\ with A000043(n)=[...][n], the dots being replaced by DATA from A000043. - M. F. Hasler, Dec 05 2018

Formula

a(n) = A000005(A000396(n)).

a(n) = floor{log_2(A000396(n))} + 2. - Lekraj Beedassy, Aug 21 2004

a(n) = 2*A000043(n). - M. F. Hasler, Dec 05 2018

Extensions

Definition changed (inserting the word "even") by Ivan Panchenko, Apr 16 2018

a(38)-a(39) from Ivan Panchenko, Apr 16 2018