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

Author

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

Keywords

• nonn

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.

Crossrefs

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

Programs

Mathematica

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

Formula

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

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

Extensions

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