A133028 Even perfect numbers divided by 2.

3, 14, 248, 4064, 16775168, 4294934528, 68719345664, 1152921504069976064, 1329227995784915872327346307976921088, 95780971304118053647396689042151819065498660774084608, 6582018229284824168619876730229361455111736159193471558891864064, 7237005577332262213973186563042994240786838745737417944533177174565599576064

Offset: 1

Author

Omar E. Pol, Oct 20 2007, Apr 23 2008, Apr 28 2009

Keywords

• nonn

Comments

a(13) has 314 digits and is too large to include. - R. J. Mathar, Oct 23 2007

Largest proper divisor of n-th even perfect number.

Also numbers k such that A000203(k) is divisible 24. - Ctibor O. Zizka, Jun 29 2009

Links

• Amiram Eldar, Table of n, a(n) for n = 1..15
• Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Crossrefs

Cf. A028334, A000396 (perfect numbers).

Cf. A000043, A000668, A000203.

Cf. A018254, A018487, A032742, A133024, A133025, A135652, A135653, A135654, A135655.

Cf. A134708, A135650. - Omar E. Pol, Jul 07 2009

Programs

Maple

a:=proc(n) if isprime(2^n-1)=true then 2^(n-2)*(2^n-1) else end if end proc: seq(a(n),n=1..120); # Emeric Deutsch, Oct 24 2007

Mathematica

p = Select[2^Range[400] - 1, PrimeQ]; p*(p+1)/4 (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
Map[2^(#-2) * (2^# - 1) &, MersennePrimeExponent[Range[12]]] (* Amiram Eldar, Oct 21 2024 *)

Formula

a(n) = A000396(n)/2. - R. J. Mathar, Oct 23 2007 [Assuming there are no odd perfect numbers. - Jianing Song, Sep 17 2022]

a(n) = 2^(A000043(n) - 2) * A000668(n). - Omar E. Pol, Mar 01 2008

a(n) = A032742(A000396(n)), assuming there are no odd perfect numbers.

Extensions

More terms from R. J. Mathar and Emeric Deutsch, Oct 23 2007