A061645 -id:A061645 - 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.
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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)
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P. G. Brown, A Note on Ramanujan's (FALSE) Conjectures Regarding 'Mersenne Primes'
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C. K. Caldwell, Recent Mersenne primes
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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
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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

A000396 Perfect numbers k: k is equal to the sum of the proper divisors of k.

Original entry on oeis.org

6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216

Offset: 1

N. J. A. Sloane

A number k is abundant if sigma(k) > 2k (cf. A005101), perfect if sigma(k) = 2k (this sequence), or deficient if sigma(k) < 2k (cf. A005100), where sigma(k) is the sum of the divisors of k (A000203).
The numbers 2^(p-1)*(2^p - 1) are perfect, where p is a prime such that 2^p - 1 is also prime (for the list of p's see A000043). There are no other even perfect numbers and it is believed that there are no odd perfect numbers.
Numbers k such that Sum_{d|k} 1/d = 2. - Benoit Cloitre, Apr 07 2002
For number of divisors of a(n) see A061645(n). Number of digits in a(n) is A061193(n). - Lekraj Beedassy, Jun 04 2004
All terms other than the first have digital root 1 (since 4^2 == 4 (mod 6), we have, by induction, 4^k == 4 (mod 6), or 2*2^(2*k) = 8 == 2 (mod 6), implying that Mersenne primes M = 2^p - 1, for odd p, are of the form 6*t+1). Thus perfect numbers N, being M-th triangular, have the form (6*t+1)*(3*t+1), whence the property N mod 9 = 1 for all N after the first. - Lekraj Beedassy, Aug 21 2004
The earliest recorded mention of this sequence is in Euclid's Elements, IX 36, about 300 BC. - Artur Jasinski, Jan 25 2006
Theorem (Euclid, Euler). An even number m is a perfect number if and only if m = 2^(k-1)*(2^k-1), where 2^k-1 is prime. Euler's idea came from Euclid's Proposition 36 of Book IX (see Weil). It follows that every even perfect number is also a triangular number. - Mohammad K. Azarian, Apr 16 2008
Triangular numbers (also generalized hexagonal numbers) A000217 whose indices are Mersenne primes A000668, assuming there are no odd perfect numbers. - Omar E. Pol, May 09 2008, Sep 15 2013
If a(n) is even, then 2*a(n) is in A181595. - Vladimir Shevelev, Nov 07 2010
Except for a(1) = 6, all even terms are of the form 30*k - 2 or 45*k + 1. - Arkadiusz Wesolowski, Mar 11 2012
a(4) = A229381(1) = 8128 is the "Simpsons's perfect number". - Jonathan Sondow, Jan 02 2015
Theorem (Farideh Firoozbakht): If m is an integer and both p and p^k-m-1 are prime numbers then x = p^(k-1)*(p^k-m-1) is a solution to the equation sigma(x) = (p*x+m)/(p-1). For example, if we take m=0 and p=2 we get Euclid's result about perfect numbers. - Farideh Firoozbakht, Mar 01 2015
The cototient of the even perfect numbers is a square; in particular, if 2^p - 1 is a Mersenne prime, cototient(2^(p-1) * (2^p - 1)) = (2^(p-1))^2 (see A152921). So, this sequence is a subsequence of A063752. - Bernard Schott, Jan 11 2019
Euler's (1747) proof that all the even perfect number are of the form 2^(p-1)*(2^p-1) implies that their asymptotic density is 0. Kanold (1954) proved that the asymptotic density of odd perfect numbers is 0. - Amiram Eldar, Feb 13 2021
If k is perfect and semiprime, then k = 6. - Alexandra Hercilia Pereira Silva, Aug 30 2021
This sequence lists the fixed points of A001065. - Alois P. Heinz, Mar 10 2024

			6 is perfect because 6 = 1+2+3, the sum of all divisors of 6 less than 6; 28 is perfect because 28 = 1+2+4+7+14.

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Index entries for "core" sequences
Index entries for sequences where any odd perfect numbers must occur

See A000043 for the current state of knowledge about Mersenne primes.
Cf. A007539, A005820, A027687, A046060, A046061, A000668, A090748, A133033, A000217, A000384, A019279, A061652, A006516, A144912, A153800, A007593, A220290, A028499-A028502, A034916, A065549, A275496, A063752, A156552, A152921, A324201.
Cf. A228058 for Euler's criterion for odd terms.
Positions of 0's in A033879 and in A033880.
Subsequence of following sequences: A005835, A006039, A007691, A023196, A043305, A065997, A083207, A109510, A118372, A216782, A246282, A263837, A294900, A333646, A334410, A335267, A336702, A341622, A342922, A344755, A352739, A357462, and (the even terms), of: A005153, A063752, A174973, A336547, A338520.
Cf. A001065.

a000396 n = a000396_list !! (n-1)
a000396_list = [x | x <- [1..], a000203 x == 2 * x]
-- Reinhard Zumkeller, Jan 20 2012

Select[Range[9000], DivisorSigma[1,#]== 2*# &] (* G. C. Greubel, Oct 03 2017 *)
PerfectNumber[Range[15]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 10 2018 *)

PARI
```
isA000396(n) = (sigma(n) == 2*n);
```

from sympy import divisor_sigma
def ok(n): return n > 0 and divisor_sigma(n) == 2*n
print([k for k in range(9999) if ok(k)]) # Michael S. Branicky, Mar 12 2022

The perfect number N = 2^(p-1)*(2^p - 1) is also multiplicatively p-perfect (i.e., A007955(N) = N^p), since tau(N) = 2*p. - Lekraj Beedassy, Sep 21 2004
a(n) = 2^A133033(n) - 2^A090748(n), assuming there are no odd perfect numbers. - Omar E. Pol, Feb 28 2008
a(n) = A000668(n)*(A000668(n)+1)/2, assuming there are no odd perfect numbers. - Omar E. Pol, Apr 23 2008
a(n) = A000217(A000668(n)), assuming there are no odd perfect numbers. - Omar E. Pol, May 09 2008
a(n) = Sum of the first A000668(n) positive integers, assuming there are no odd perfect numbers. - Omar E. Pol, May 09 2008
a(n) = A000384(A019279(n)), assuming there are no odd perfect numbers and no odd superperfect numbers. a(n) = A000384(A061652(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Aug 17 2008
a(n) = A006516(A000043(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Aug 30 2008
From Reikku Kulon, Oct 14 2008: (Start)
A144912(2, a(n)) = 1;
A144912(4, a(n)) = -1 for n > 1;
A144912(8, a(n)) = 5 or -5 for all n except 2;
A144912(16, a(n)) = -4 or -13 for n > 1. (End)
a(n) = A019279(n)*A000668(n), assuming there are no odd perfect numbers and odd superperfect numbers. a(n) = A061652(n)*A000668(n), assuming there are no odd perfect numbers. - Omar E. Pol, Jan 09 2009
a(n) = A007691(A153800(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Jan 14 2009
Even perfect numbers N = K*A000203(K), where K = A019279(n) = 2^(p-1), A000203(A019279(n)) = A000668(n) = 2^p - 1 = M(p), p = A000043(n). - Lekraj Beedassy, May 02 2009
a(n) = A060286(A016027(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Dec 13 2012
For n >= 2, a(n) = Sum_{k=1..A065549(n)} (2*k-1)^3, assuming there are no odd perfect numbers. - Derek Orr, Sep 28 2013
a(n) = A275496(2^((A000043(n) - 1)/2)) - 2^A000043(n), assuming there are no odd perfect numbers. - Daniel Poveda Parrilla, Aug 16 2016
a(n) = A156552(A324201(n)), assuming there are no odd perfect numbers. - Antti Karttunen, Mar 28 2019
a(n) = ((2^(A000043(n)))^3 - (2^(A000043(n)) - 1)^3 - 1)/6, assuming there are no odd perfect numbers. - Jules Beauchamp, Jun 06 2025

I removed a large number of comments that assumed there are no odd perfect numbers. There were so many it was getting hard to tell which comments were true and which were conjectures. - N. J. A. Sloane, Apr 16 2023

Reference to Albert H. Beiler's book updated by Harvey P. Dale, Jan 13 2025

A111592 Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n.

Original entry on oeis.org

12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222, 224, 234, 246, 258, 270, 282, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 650, 654, 672, 678, 762, 786, 812

Offset: 1

Jason Earls, Aug 09 2005

All admirable numbers are abundant.
If 2^n-2^k-1 is an odd prime then m=2^(n-1)*(2^n-2^k-1) is in the sequence because 2^k is one of the proper divisors of m and sigma(m)-2m=(2^n-1)*(2^n-2^k)-2^n*(2^n-2^k-1)=2^k hence m=(sigma(m)-m)-2^k, namely m is an Admirable number. This is one of the results of the following theorem that I have found. Theorem: If 2^n-j-1 is an odd prime and m=2^(n-1)*(2^n-j-1) then sigma(m)-2m=j. The case j=0 is well known. - Farideh Firoozbakht, Jan 28 2006
In particular, these numbers have abundancy 2 to 3: 2 < sigma(n)/n <= 3. - Charles R Greathouse IV, Jan 30 2014
Subsequence of A083207. - Ivan N. Ianakiev, Mar 20 2017
The concept of admirable numbers was developed by educator Jerome Michael Sachs (1914-2012) for a television in-service training course in mathematics for elementary school teachers. - Amiram Eldar, Aug 22 2018
Odd terms are listed in A109729. For abundant nonsquares, it is equivalent to say sigma(n)/2 - n divides n. For squares, sigma(n)/2 - n is half-integer, but n could still be an integer multiple. This first occurs for n = m^2 with even m = 2^k*(2^(2*k+1)-1), k = 1, 2, 3, 6, ... (A146768), and odd m = 13167. - M. F. Hasler, Jan 26 2020

			12 = 1+3+4+6-2, 20 = 2+4+5+10-1, etc.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Giovanni Resta, admirable numbers
J. M. Sachs, Admirable Numbers and Compatible Pairs, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
T. Trotter, Admirable Numbers

Subsequence of A005101 (abundant numbers).
Cf. A000396 (perfect numbers), A005100 (deficient numbers), A000203 (sigma), A061645.
Cf. A109729 (odd admirable numbers).

with(numtheory); isadmirable := proc(n) local b, d, S; b:=false; S:=divisors(n) minus {n}; for d in S do if sigma(n)-2*d=2*n then b:=true; break fi od; return b; end: select(proc(z) isadmirable(z) end, [$1..1000]); # Walter Kehowski, Aug 12 2005

fQ[n_] := Block[{d = Most[Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; Select[ Range[821], fQ[ # ] &] (* Robert G. Wilson v, Aug 13 2005 *)
Select[Range[812],MemberQ[Most[Divisors[#]],(DivisorSigma[1,#]-2*#)/2]&] (* Ivan N. Ianakiev, Mar 23 2017 *)

for(n=1,10^3,ap=sigma(n)-2*n;if(ap>0 && (ap%2)==0,d=ap/2;if(d!=n && (n%d)==0, print1(n",")))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

is(n)=if(issquare(n)||issquare(n/2),0,my(d=sigma(n)/2-n); d>0 && d!=n && n%d==0) \\ Charles R Greathouse IV, Jun 21 2011

Better definition from Walter Kehowski, Aug 12 2005

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

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

Original entry on oeis.org

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

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

nonn

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

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.

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

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

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

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

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

A135654 Divisors of 8128 (the 4th perfect number), written in base 2.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 100000, 1000000, 1111111, 11111110, 111111100, 1111111000, 11111110000, 111111100000, 1111111000000

Offset: 1

Omar E. Pol, Feb 23 2008, Mar 03 2008

The number of divisors of the 4th perfect number is equal to 2*A000043(4)=A061645(4)=14.

			The structure of divisors of 8128 (see A133024)
-------------------------------------------------------------------------
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)........ 32 = 2^5 ........... 100000
7)........ 64 = 2^6 ........... 1000000 ... (The 4th superperfect number)
8)....... 127 = 2^7 - 2^0 ..... 1111111 ... (The 4th Mersenne prime)
9)....... 254 = 2^8 - 2^1 ..... 11111110
10)...... 508 = 2^9 - 2^2 ..... 111111100
11)..... 1016 = 2^10- 2^3 ..... 1111111000
12)..... 2032 = 2^11- 2^4 ..... 11111110000
13)..... 4064 = 2^12- 2^5 ..... 111111100000
14)..... 8128 = 2^13- 2^6 ..... 1111111000000 ... (The 4th perfect number)

Index entries for sequences related to divisors of numbers

For more information see A133024 (Divisors of 8128). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

FromDigits[IntegerDigits[#,2]]&/@Divisors[8128] (* Harvey P. Dale, Jan 08 2014 *)

a(n)=A133024(n), written in base 2. Also, for n=1 .. 14: If n<=(A000043(4)=7) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(4)=7 digits "1" and (n-1-A000043(4)) digits "0".

A135655 Divisors of 33550336 (the 5th perfect number), written in base 2.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 1111111111111, 11111111111110, 111111111111100, 1111111111111000, 11111111111110000

Offset: 1

Omar E. Pol, Feb 23 2008, Mar 01 2008, Mar 03 2008

The number of divisors of the 5th perfect number is equal to 2*A000043(5)=A061645(5)=26.

			The structure of divisors of 33550336 (see A133025)
------------------------------------------------------------------------
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)........... 32 = 2^5 ........... 100000
7)........... 64 = 2^6 ........... 1000000
8).......... 128 = 2^7 ........... 10000000
9).......... 256 = 2^8 ........... 100000000
10)......... 512 = 2^9 ........... 1000000000
11)........ 1024 = 2^10 .......... 10000000000
12)........ 2048 = 2^11 .......... 100000000000
13) ....... 4096 = 2^12 .......... 1000000000000 ... (The 5th superperfect number)
14) ....... 8191 = 2^13 - 2^0 .... 1111111111111 ... (The 5th Mersenne prime)
15) ...... 16382 = 2^14 - 2^1 .... 11111111111110
16) ...... 32764 = 2^15 - 2^2 .... 111111111111100
17) ...... 65528 = 2^16 - 2^3 .... 1111111111111000
18) ..... 131056 = 2^17 - 2^4 .... 11111111111110000
19) ..... 262112 = 2^18 - 2^5 .... 111111111111100000
20) ..... 524224 = 2^19 - 2^6 .... 1111111111111000000
21) .... 1048448 = 2^20 - 2^7 .... 11111111111110000000
22) .... 2096896 = 2^21 - 2^8 .... 111111111111100000000
23) .... 4193792 = 2^22 - 2^9 .... 1111111111111000000000
24) .... 8387584 = 2^23 - 2^10 ... 11111111111110000000000
25) ... 16775168 = 2^24 - 2^11 ... 111111111111100000000000
26) ... 33550336 = 2^25 - 2^12 ... 1111111111111000000000000 ... (The 5th perfect number)

Omar E. Pol, Table of n, a(n) for n = 1..26 (shows all terms).
Index entries for sequences related to divisors of numbers

For more information see A133025 (Divisors of 33550336). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

a(n)=A133025(n), written in base 2. Also, for n=1 .. 26: If n<=(A000043(5)=13) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(5)=13 digits "1" and (n-1-A000043(5)) digits "0".

A135650 Even perfect numbers written in base 2.

Original entry on oeis.org

110, 11100, 111110000, 1111111000000, 1111111111111000000000000, 111111111111111110000000000000000, 1111111111111111111000000000000000000

Offset: 1

Omar E. Pol, Feb 21 2008, Feb 22 2008, Apr 28 2009

The number of digits of a(n) is equal to 2*A000043(n)-1. The central digit is "1". The first digits are "1". The last digits are "0". The number of digits "1" is equal A000043(n). The number of digits "0" is equal A000043(n)-1.
The concatenation of digits "1" of a(n) gives the n-th Mersenne prime written in binary (see A117293(n)).
Also, the number of digits of a(n) is equal to A133033(n), the number of proper divisors of n-th even perfect number.

			a(3) = 111110000 because the 3rd even perfect number is 496 and 496 written in base 2 is 111110000. Note that 11111 is the 3rd Mersenne prime A000668(3) = 31 written in base 2.

Amiram Eldar, Table of n, a(n) for n = 1..12
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A000668, A090748, A117293.

Cf. A061645, A133033.

Map[FromDigits[IntegerDigits[#, 2]] &, PerfectNumber[Range[8], "Even"]] (* Amiram Eldar, Oct 18 2024 *)

A135652 Divisors of 28 (the 2nd perfect number), written in base 2.

Original entry on oeis.org

1, 10, 100, 111, 1110, 11100

Offset: 1

Omar E. Pol, Feb 23 2008, Mar 03 2008

The number of divisors of the second perfect number is equal to 2*A000043(2)=A061645(2)=6.

			The structure of divisors of 28 (see A018254)
----------------------------------------------------------------------
n ... Divisor . Formula ....... Divisor written in base 2 ............
----------------------------------------------------------------------
1)......... 1 = 2^0 ........... 1
2)......... 2 = 2^1 ........... 10
3)......... 4 = 2^2 ........... 100 .... (The 2nd superperfect number)
4)......... 7 = 2^3 - 2^0 ..... 111 .... (The 2nd Mersenne prime)
5)........ 14 = 2^4 - 2^1 ..... 1110
6)........ 28 = 2^5 - 2^2 ..... 11100... (The 2nd perfect number)

Index entries for sequences related to divisors of numbers

For more information see A018254 (Divisors of 28). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

FromDigits[IntegerDigits[#,2]]&/@Divisors[28] (* Harvey P. Dale, Nov 14 2020 *)

apply(n->fromdigits(binary(n)), divisors(28)) \\ Charles R Greathouse IV, Jun 21 2017

a(n)=A018254(n), written in base 2. Also, for n=1 .. 6: If n<=(A000043(2)=3) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(2)=3 digits "1" and (n-1-A000043(2)) digits "0".

A135653 Divisors of 496 (the 3rd perfect number), written in base 2.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 11111, 111110, 1111100, 11111000, 111110000

Offset: 1

Omar E. Pol, Feb 23 2008, Mar 03 2008

The number of divisors of the third perfect number is equal to 2*A000043(3)=A061645(3)=10.

			The structure of divisors of 496 (see A018487)
-------------------------------------------------------------------------
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 ... (The 3rd superperfect number)
6)........ 31 = 2^5 - 2^0 ..... 11111 ... (The 3rd Mersenne prime)
7)........ 62 = 2^6 - 2^1 ..... 111110
8)....... 124 = 2^7 - 2^2 ..... 1111100
9)....... 248 = 2^8 - 2^3 ..... 11111000
10)...... 496 = 2^9 - 2^4 ..... 111110000 ... (The 3rd perfect number)

Index entries for sequences related to divisors of numbers

For more information see A018487 (Divisors of 496). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

FromDigits[IntegerDigits[#,2]]&/@Divisors[496] (* Harvey P. Dale, Dec 02 2018 *)

apply(n->fromdigits(binary(n)), divisors(496)) \\ Charles R Greathouse IV, Jun 21 2017

a(n)=A018487(n), written in base 2. Also, for n=1 .. 10: If n<=(A000043(3)=5) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(3)=5 digits "1" and (n-1-A000043(3)) digits "0".

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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A000396 Perfect numbers k: k is equal to the sum of the proper divisors of k.

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A111592 Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n.

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

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

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A135654 Divisors of 8128 (the 4th perfect number), written in base 2.

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