A061645 - OEIS

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

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OFFSET

1,1

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

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).

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

EXAMPLE

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

MATHEMATICA

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

PROG