A133033 Number of proper divisors of n-th even perfect number.

3, 5, 9, 13, 25, 33, 37, 61, 121, 177, 213, 253, 1041, 1213, 2557, 4405, 4561, 6433, 8505, 8845, 19377, 19881, 22425, 39873, 43401, 46417, 88993, 172485, 221005, 264097, 432181, 1513677, 1718865, 2515573, 2796537, 5952441, 6042753, 13945185, 26933833, 41992021, 48073165, 51929901, 60804913

Offset

1,1

Comments

Perfect numbers: A000396(n) = 2^a(n) - 2^A090748(n), assuming there are no odd perfect numbers.

Also, a(n) is equal to the number of bits in A135650(n), the n-th even perfect number written in base 2.

These are odd numbers k such that 2^((k+1)/2) - 1 is (a Mersenne) prime. - Thomas Ordowski, Apr 20 2019

Links

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)

Chris K. Caldwell, "Top Twenty" page, Mersenne Primes.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Formula

a(n) = A061645(n) - 1.

a(n) = A000043(n) + A090748(n) = 2*A000043(n) - 1 = 2*A090748(n) + 1.

Mathematica

2 * MersennePrimeExponent[Range[48]] - 1 (* Amiram Eldar, Oct 18 2024 *)

Crossrefs

Cf. A000043, A000396, A061645, A090748, A135650.

Sequence in context: A153263 A295140 A144933 * A145365 A306833 A146433

Adjacent sequences: A133030 A133031 A133032 * A133034 A133035 A133036

Keyword

nonn

Author

Omar E. Pol, Oct 27 2007, Feb 23 2008, Apr 28 2009

Extensions

a(39)-a(43) from Ivan Panchenko, Apr 12 2018

Status

approved