A135654 -id:A135654 - cp's OEIS frontend

A000043 Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281

Offset: 1

N. J. A. Sloane

Equivalently, integers k such that 2^k - 1 is prime.
It is believed (but unproved) that this sequence is infinite. The data suggest that the number of terms up to exponent N is roughly K log N for some constant K.
Length of prime repunits in base 2.
The associated perfect number N=2^(p-1)*M(p) (=A019279*A000668=A000396), has 2p (=A061645) divisors with harmonic mean p (and geometric mean sqrt(N)). - Lekraj Beedassy, Aug 21 2004
In one of his first publications Euler found the numbers up to 31 but erroneously included 41 and 47.
Equals number of bits in binary expansion of n-th Mersenne prime (A117293). - Artur Jasinski, Feb 09 2007
Number of divisors of n-th even perfect number, divided by 2. Number of divisors of n-th even perfect number that are powers of 2. Number of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n). - Omar E. Pol, Feb 24 2008
Number of divisors of n-th even superperfect number A061652(n). Numbers of divisors of n-th superperfect number A019279(n), assuming there are no odd superperfect numbers. - Omar E. Pol, Mar 01 2008
Differences between exponents when the even perfect numbers are represented as differences of powers of 2, for example: The 5th even perfect number is 33550336 = 2^25 - 2^12 then a(5)=25-12=13 (see A135655, A133033, A090748). - Omar E. Pol, Mar 01 2008
Number of 1's in binary expansion of n-th even perfect number (see A135650). Number of 1's in binary expansion of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n) (see A135652, A135653, A135654, A135655). - Omar E. Pol, May 04 2008
Indices of the numbers A006516 that are also even perfect numbers. - Omar E. Pol, Aug 30 2008
Indices of Mersenne numbers A000225 that are also Mersenne primes A000668. - Omar E. Pol, Aug 31 2008
The (prime) number p appears in this sequence if and only if there is no prime q<2^p-1 such that the order of 2 modulo q equals p; a special case is that if p=4k+3 is prime and also q=2p+1 is prime then the order of 2 modulo q is p so p is not a term of this sequence. - Joerg Arndt, Jan 16 2011
Primes p such that sigma(2^p) - sigma(2^p-1) = 2^p-1. - Jaroslav Krizek, Aug 02 2013
Integers k such that every degree k irreducible polynomial over GF(2) is also primitive, i.e., has order 2^k-1.  Equivalently, the integers k such that A001037(k) = A011260(k). - Geoffrey Critzer, Dec 08 2019
Conjecture: for k > 1, 2^k-1 is (a Mersenne) prime or k = 2^(2^m)+1 (is a Fermat number) if and only if (k-1)^(2^k-2) == 1 (mod (2^k-1)k^2). - Thomas Ordowski, Oct 05 2023
Conjecture: for p prime, 2^p-1 is (a Mersenne) prime or p = 2^(2^m)+1 (is a Fermat number) if and only if (p-1)^(2^p-2) == 1 (mod 2^p-1). - David Barina, Nov 25 2024
Already as of Dec. 2020, all exponents up to 10^8 had been verified, implying that 74207281, 77232917 and 82589933 are indeed the next three terms. As of today, all exponents up to 130439863 have been tested at least once, see the GIMPS Milestones Report. - M. F. Hasler, Apr 11 2025
On June 23. 2025 all exponents up to 74340751 have been verified, confirming that 74207281 is the exponent of the 49th Mersenne Prime. - Rodolfo Ruiz-Huidobro, Jun 23 2025

			Corresponding to the initial terms 2, 3, 5, 7, 13, 17, 19, 31 ... we get the Mersenne primes 2^2 - 1 = 3, 2^3 - 1 = 7, 2^5 - 1 = 31, 127, 8191, 131071, 524287, 2147483647, ... (see A000668).

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.
J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 79.
R. K. Guy, Unsolved Problems in Number Theory, Section A3.
F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 57.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 19.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 132-134.
B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 1971), Abstract 684-A15, p. 608.

P. T. Bateman, J. L. Selfridge, and S. S. Wagstaff, Jr., The new Mersenne conjecture, Amer. Math. Monthly 96 (1989), no. 2, 125--128. MR0992073 (90c:11009).
J. Bernheiden, Mersenne Numbers (Text in German)
Andrew R. Booker, The Nth Prime Page
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
P. G. Brown, A Note on Ramanujan's (FALSE) Conjectures Regarding 'Mersenne Primes'
C. K. Caldwell, Mersenne Primes
C. K. Caldwell, Recent Mersenne primes
Zuling Chang, Martianus Frederic Ezerman, Adamas Aqsa, Fahreza, San Ling, Janusz Szmidt, and Huaxiong Wang, Binary de Bruijn Sequences via Zech's Logarithms, 2018.
Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
Leonhard Euler, Observations on a theorem of Fermat and others on looking at prime numbers, arXiv:math/0501118 [math.HO], 2005-2008.
Leonhard Euler, Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus
G. Everest et al., Primes generated by recurrence sequences, arXiv:math/0412079 [math.NT], 2006.
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Luis H. Gallardo and Olivier Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.
Donald B. Gillies, Three new Mersenne primes and a statistical theory Mathematics of Computation 18.85 (1964): 93-97.
GIMPS (Great Internet Mersenne Prime Search), Distributed Computing Projects
GIMPS, Milestones Report
GIMPS, GIMPS Project discovers largest known prime number 2^77232917-1
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
H. Lifchitz, Mersenne and Fermat primes field
A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see p. 143.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
Romeo Meštrović, Goldbach-type conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
G. P. Michon, Perfect Numbers, Mersenne Primes
Albert A. Mullin, Letter to the editor, about "The new Mersenne conjecture" [Amer. Math. Monthly 96 (1989), no. 2, 125-128; MR0992073 (90c:11009)] by P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., Amer. Math. Monthly 96 (1989), no. 6, 511. MR0999415 (90f:11008).
Curt Noll and Laura Nickel, The 25th and 26th Mersenne primes, Math. Comp. 35 (1980), 1387-1390.
M. Oakes, A new series of Mersenne-like Gaussian primes
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
H. J. Smith, Mersenne Primes
B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 (1971), 2319-2320.
H. S. Uhler, On All Of Mersenne's Numbers Particularly M_193, PNAS 1948 34 (3) 102-103.
H. S. Uhler, First Proof That The Mersenne Number M_157 Is Composite, PNAS 1944 30(10) 314-316.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Cunningham Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Mersenne Prime
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Wagstaff's Conjecture
David Whitehouse, Number takes prime position (2^13466917 - 1 found after 13000 years of computer time)
K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte für Mathematik und Physik, Vol. 3, No. 1 (1892), 265-284.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime
Index entries for "core" sequences

Cf. A000668 (Mersenne primes).
Cf. A028335 (integer lengths of Mersenne primes).
Cf. A000225 (Mersenne numbers).
Cf. A001348 (Mersenne numbers with n prime).
Cf. A016027, A046051, A057429, A057951-A057958, A066408, A117293, A127962, A127963, A127964, A127965, A127961, A000979, A000978, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A134458, A000225, A000396, A090748, A133033, A135655, A006516, A019279, A061652, A133033, A135650, A135652, A135653, A135654, A260073, A050475, A379590.

MersennePrimeExponent[Range[48]] (* Eric W. Weisstein, Jul 17 2017; updated Oct 21 2024 *)

isA000043(n) = isprime(2^n-1) \\ Michael B. Porter, Oct 28 2009

is(n)=my(h=Mod(2,2^n-1)); for(i=1, n-2, h=2*h^2-1); h==0||n==2 \\ Lucas-Lehmer test for exponent e. - Joerg Arndt, Jan 16 2011, and Charles R Greathouse IV, Jun 05 2013
forprime(e=2,5000,if(is(e),print1(e,", "))); /* terms < 5000 */

from sympy import isprime, prime
for n in range(1,100):
    if isprime(2**prime(n)-1):
        print(prime(n), end=', ') # Stefano Spezia, Dec 06 2018

a(n) = log((1/2)*(1+sqrt(1+8*A000396(n))))/log(2). - Artur Jasinski, Sep 23 2008 (under the assumption there are no odd perfect numbers, Joerg Arndt, Feb 23 2014)
a(n) = A000005(A061652(n)). - Omar E. Pol, Aug 26 2009
a(n) = A000120(A000396(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Oct 30 2013

Also in the sequence: p = 74207281. - Charles R Greathouse IV, Jan 19 2016
Also in the sequence: p = 77232917. - Eric W. Weisstein, Jan 03 2018
Also in the sequence: p = 82589933. - Gord Palameta, Dec 21 2018
a(46) = 42643801 and a(47) = 43112609, whose ordinal positions in the sequence are now confirmed, communicated by Eric W. Weisstein, Apr 12 2018
a(48) = 57885161, whose ordinal position in the sequence is now confirmed, communicated by Benjamin Przybocki, Jan 05 2022
Also in the sequence: p = 136279841. - Eric W. Weisstein, Oct 21 2024
As of Jan 31 2025, 48 terms are known, and are shown in the DATA section. Four additional numbers are known to be in the sequence, namely 74207281, 77232917, 82589933, and 136279841, but they may not be the next terms. See the GIMP website for the latest information. - N. J. A. Sloane, Jan 31 2025

A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).

Original entry on oeis.org

8, 32, 512, 8192, 33554432, 8589934592, 137438953472, 2305843009213693952, 2658455991569831745807614120560689152, 191561942608236107294793378393788647952342390272950272

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Sum of n-th even perfect number and n-th even superperfect number.

Also, sum of n-th perfect number and n-th superperfect number, if there are no odd perfect and odd superperfect numbers, then the n-th perfect number is the difference between a(n) and the n-th superperfect number (see A135652, A135653, A135654 and A135655).

			a(5) = 33554432 because A000043(5) = 13 and 2^(2*13 - 1) = 2^25 = 33554432.
Also, if there are no odd perfect and odd superperfect numbers then we can write a(5) = A000396(5) + A019279(5) = A000396(5) + A061652(5) = 33554432.

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

Cf. A000079, A000396, A019279, A061645, A061652, A133033, A135652, A135653, A135654, A135655, A139286, A139294, A139307.

Cf. A000043, A000668, A072868.

2^(2 * MersennePrimeExponent[Range[10]] - 1) (* Amiram Eldar, Oct 17 2024 *)

a(n) = 2^(2*A000043(n) - 1). Also, a(n) = 2^A133033(n), if there are no odd perfect numbers. Also, a(n) = A000396(n) + A019279(n), if there are no odd perfect and odd superperfect numbers. Also, a(n) = A000396(n) + A061652(n), if there are no odd perfect numbers, then we can write: perfect number A000396(n) = a(n) - A061652(n).

a(n) = A061652(n)*(A000668(n)+1) = A061652(n)*A072868(n). - Omar E. Pol, Apr 13 2008

A133028 Even perfect numbers divided by 2.

Original entry on oeis.org

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

Offset: 1

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

nonn

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

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

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

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

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 *)

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.

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

A133049 Squares of Mersenne primes A000668(n).

Original entry on oeis.org

9, 49, 961, 16129, 67092481, 17179607041, 274876858369, 4611686014132420609, 5316911983139663487003542222693990401, 383123885216472214589586755549637256619304505646776321

Offset: 1

Omar E. Pol, Oct 30 2007, Apr 23 2008

nonn

Sum of last A000043(n) divisors of the n-th even perfect number. In other words; sum of divisors that are not powers of 2 of the n-th even perfect number, or sum of divisors that are multiples of the n-th Mersenne prime A000668(n) of the n-th even perfect number. See A139247 for more information.

See the structure of the divisors of perfect numbers in A135652, A135653, A135654 and A135655.

			a(3)=961 because the 3rd Mersenne prime is 31 and 31^2=961.

G. C. Greubel, Table of n, a(n) for n = 1..15
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000290, A001248. Mersenne primes: A000668.

Cf. A000043, A000396, A135652, A135653, A135654, A135655, A138247.

Select[2^Range[1000] - 1, PrimeQ]^2 (* G. C. Greubel, Oct 03 2017 *)

forprime(p=2, 1000, if(ispseudoprime(2^p-1), print1((2^p-1)^2", "))) \\ G. C. Greubel, Oct 03 2017

a(n) = A000668(n)^2

More terms from Olaf Voß, Feb 13 2008

A133024 Divisors of 8128, the 4th perfect number.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128

Offset: 1

Omar E. Pol, Oct 26 2007, Mar 03 2008, Dec 27 2008

127 is the 4th Mersenne prime: A000668.
The number of divisors of the 4th perfect number is 2*A000043(4)=A061645(4)=14.
For the structure of this sequence and its binary expansion, see A135654.

Cf. A018254, A018487. Perfect numbers: A000396.

Cf. A000079, A000668, A061645, A133033, A135654.

Divisors[8128] (* Harvey P. Dale, Jul 30 2023 *)

divisors(8128) \\ Charles R Greathouse IV, Jun 21 2017

For n=1..7 : a(n) = 2^(n-1). For n=8..14: a(n) = 2^(n-1) - 2^(n-8) = A000668(4)*2^(n-8).

A139247 Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 6, 7, 14, 28, 31, 62, 124, 248, 496, 127, 254, 508, 1016, 2032, 4064, 8128, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 33550336, 131071, 262142, 524284, 1048568, 2097136, 4193792

Offset: 1

Omar E. Pol, Apr 22 2008

Also, row n list the divisors of n-th perfect number that are not powers of 2.

First term of row n is the n-th Mersenne prime A000668(n). Last term of row n is the n-th perfect number A000396(n). Row n has A000043(n) terms. The sum of row n is equal to A133049(n), the square of n-th Mersenne prime A000668(n).

			Triangle begins:
  3, 6,
  7, 14, 28
  31, 62, 124, 248, 496
  127, 254, 508, 1016, 2032, 4064, 8128
  ...
==========================================================
Row .... First term ..... Last term ....... Row sum ......
n ..... (A000668(n)) ... (A000396(n)) ... (A000668(n)^2) .
==========================================================
1 ............ 3 .............. 6 ......... 3^2 = 9
2 ............ 7 ............. 28 ......... 7^2 = 49
3 ........... 31 ............ 496 ........ 31^2 = 961
4 .......... 127 ........... 8128 ....... 127^2 = 16129
5 ......... 8191 ....... 33550336 ...... 8191^2 = 67092481

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

Cf. A000043, A000396, A000668, A018254, A018487, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655.

A035526 Reverse and add (in binary).

Original entry on oeis.org

1, 10, 11, 110, 1001, 10010, 11011, 110110, 1010001, 10010110, 11111111, 111111110, 1011111101, 10111111010, 100011110111, 1011111101000, 1101011100101, 10111111010000, 11001011001101, 101111110100000

Offset: 0

N. J. A. Sloane, E. M. Rains

First 4 members are the divisors of 6 (the first perfect number), written in base 2 (see A135652, A135653, A135654, A135655). - Omar E. Pol, May 04 2008

			110 + 011 = 1001.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A035522 for same sequence but written in base 10.

Cf. A007088.

a035526 = a007088 . a035522  -- Reinhard Zumkeller, Jan 02 2015

from itertools import accumulate, repeat
def iterate(n, _): b = str(n); return int(bin(int(b, 2)+int(b[::-1], 2))[2:])
def aupto(nn): return list(accumulate(repeat(1, nn), iterate))
print(aupto(20)) # Michael S. Branicky, Jan 10 2021

a(n) = A007088(A035522(n)). - Reinhard Zumkeller, Jan 02 2015

A094026 Expansion of x(1+10x)/((1-x^2)(1-10x^2)).

Original entry on oeis.org

0, 1, 10, 11, 110, 111, 1110, 1111, 11110, 11111, 111110, 111111, 1111110, 1111111, 11111110, 11111111, 111111110, 111111111, 1111111110, 1111111111, 11111111110, 11111111111, 111111111110, 111111111111, 1111111111110

Offset: 0

Paul Barry, Apr 22 2004

The expansion of x(1+kx)/((1-x^2)(1-kx^2)) has a(n)=k^((n+1)/2)/(2(sqrt(k)-1))-(-sqrt(k))^(n+1)/(2(sqrt(k)+1))-(-1)^n/2-(k+1)/(2(k-1)).

First 4 positive members are the divisors of 6 (the first perfect number), written in base 2 (see A135652, A135653, A135654, A135655). - Omar E. Pol, May 04 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (0,11,0,-10).

Cf. A075427, A094025, A080610.

I:=[0,1,10,11]; [n le 4 select I[n] else 11*Self(n-2)-10*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 25 2019

LinearRecurrence[{0, 11, 0, -10}, {0, 1, 10, 11}, 30] (* Vincenzo Librandi, Apr 25 2019 *)
CoefficientList[Series[x (1+10x)/((1-x^2)(1-10x^2)),{x,0,30}],x] (* Harvey P. Dale, Jul 07 2024 *)

a(n) = 10^(n/2)(5/9+sqrt(10)/18+(5/9-sqrt(10)/18)(-1)^n)-(-1)^n/2-11/18.

A138814 Divisors of 4064 (half the 4th perfect number).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 127, 254, 508, 1016, 2032, 4064

Offset: 1

Omar E. Pol, Mar 31 2008

The n-th perfect number divided by 2 (A133028(n)) has 2*A090748(n) divisors, then this sequence has 12 members. First 6 members are the first 6 powers of 2 A000079. Last 6 members are multiples of 4th Mersenne prime A000668(4)=127. a(n) written in base 2 has n digit. See A138824 for the structure of this sequence.

Index entries for sequences related to divisors of numbers

Perfect number divided by 2: A133028.

Cf. A000043, A000079, A000396, A000668, A090748, A134708, A135654, A138824.

Divisors[4064] (* Paolo Xausa, Jul 22 2024 *)

divisors(4064) \\ Charles R Greathouse IV, Jun 21 2017

A138824 Divisors of 4064 (the 4th perfect number divided by 2), written in base 2.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 100000, 1111111, 11111110, 111111100, 1111111000, 11111110000, 111111100000

Offset: 1

Omar E. Pol, Mar 31 2008

a(n) has n digits. See A138814 for more information.

			The structure of divisors of 4064 (see A138814)
.................................................................
n ........... Divisor . Formula ....... Divisor written in base 2
.................................................................
1) ................ 1 = 2^0 ........... 1
2) ................ 2 = 2^1 ........... 10
3) ................ 4 = 2^2 ........... 100
4) ................ 8 = 2^3 ........... 1000
5) ............... 16 = 2^4 ........... 10000
6) A134708(4) = .. 32 = 2^5 ........... 100000
7) A000668(4) = . 127 = 2^7 - 2^0 ..... 1111111
8) .............. 254 = 2^8 - 2^1 ..... 11111110
9) .............. 508 = 2^9 - 2^2 ..... 111111100
10) ............ 1016 = 2^10- 2^3 ..... 1111111000
11) ............ 2032 = 2^11- 2^4 ..... 11111110000
12) A133028(4) = 4064 = 2^12- 2^5 ..... 111111100000

Index entries for sequences related to divisors of numbers

Perfect number divided by 2: A133028. Cf. A000043, A000396, A000668, A090748, A134708, A135654, A138814.

FromDigits/@(IntegerDigits[#,2]&/@Divisors[4064]) (* Harvey P. Dale, Oct 12 2016 *)

cp's OEIS Frontend

A000043 Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.

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A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).

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A133028 Even perfect numbers divided by 2.

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A133049 Squares of Mersenne primes A000668(n).

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A133024 Divisors of 8128, the 4th perfect number.

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A139247 Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(n).

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A035526 Reverse and add (in binary).

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