cp's OEIS Frontend

If any odd perfect numbers exist, they should occur in this sequence, unless it is already complete, as some people believe. Please see comments and links in A005820.

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

sigma(n)/n is in A054030.

Also numbers such that the sum of the reciprocals of the divisors is an integer. - Harvey P. Dale, Jul 24 2001

Luca's solution of problem 11090, which proves that for k>1 there are an infinite number of n such that n divides sigma_k(n), does not apply to this sequence. However, it is conjectured that this sequence is also infinite. - T. D. Noe, Nov 04 2007

Numbers k such that sigma(k) is divisible by all divisors of k, subsequence of A166070. - Jaroslav Krizek, Oct 06 2009

A017666(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2012

Bach, Miller, & Shallit show that this sequence can be recognized in polynomial time with arbitrarily small error by a probabilistic Turing machine; that is, this sequence is in BPP. - Charles R Greathouse IV, Jun 21 2013

Conjecture: If n is such that 2^n-1 is in A066175 then a(n) is a triangular number. - Ivan N. Ianakiev, Aug 26 2013

Conjecture: Every multiply-perfect number is practical (A005153). I've verified this conjecture for the first 5261 terms with abundancy > 2 using Achim Flammenkamp's data. The even perfect numbers are easily shown to be practical, but every practical number > 1 is even, so a weak form says every even multiply-perfect number is practical. - Jaycob Coleman, Oct 15 2013

Numbers such that A054024(n) = 0. - Michel Marcus, Nov 16 2013

Numbers n such that k(n) = A229110(n) = antisigma(n) mod n = A024816(n) mod n = A000217(n) mod n = (n(n+1)/2) mod n = A142150(n). k(n) = n/2 for even n; k(n) = 0 for odd n (for number 1 and eventually odd multiply-perfect numbers n > 1). - Jaroslav Krizek, May 28 2014

The only terms m > 1 of this sequence that are not in A145551 are m for which sigma(m)/m is not a divisor of m. Conjecture: after 1, A323653 lists all such m (and no other numbers). - Antti Karttunen, Mar 19 2021

Let sigma^m (j) be the result of applying the sum-of-divisors function (A000203) m times to j; call j (m,k)-perfect if sigma^m (j) = k*j; then this is the sequence of (2,k)-perfect numbers.

From Michel Marcus, May 14 2016: (Start)

For these numbers, the quotient k = sigma(sigma(j))/j is an integer (see A098223). Then also k = (sigma(s)/s)*(sigma(j)/j) with s = sigma(j). That is, k = abundancy(s)*abundancy(j).

So looking at the abundancy of these terms may be interesting. Indeed we see that 459818240 and 51001180160 are actually 3-perfect numbers (A005820), and the reason they are here is that they are coprime to 3. So their sums of divisors are 4-perfect numbers (A027687), yielding q=12.

In a similar way, we can see that the 5-perfect numbers (A046060) that are coprime to 5 will be terms of this sequence with q=30. There are 20 such numbers, the smallest being 13188979363639752997731839211623940096. (End)

From Michel Marcus, May 15 2016: (Start)

It is also interesting to note that for a(2)=8, s=sigma(8)=15 is also a term. This happens to be the case for chains of several terms in a row:

8, 15, 24, 60, 168, 480 with k = 3,4,7,8,9,10;

512, 1023, 1536, 4092, 10752, 32736 with k = 3,4,7,8,9,10;

29127, 47360, 116508, 331520, 932064, 2983680 with k = 4,7,8,9,13,14;

1556480, 3932040, 14008320 with k = 9,13,14;

106151936, 251650560, 955367424 with k = 9,13,14;

312792480, 1505806848 with k = 19,20;

6604416000, 30834059256 with k = 19,20;

9623577600, 46566269568 with k = 19,20.

When j is a term, we can test if s=sigma(j) is also a term; this way we get 6 more terms: 572941926400, 845734196736, 1422976331052, 4010593484800, 11383810648416, 36095341363200.

And the corresponding chains are:

173238912000, 845734196736 with k = 19,20;

355744082763, 572941926400, 1422976331052, 4010593484800, 11383810648416, 36095341363200 with k = 4,7,8,9,13,14. (End)

From Altug Alkan, May 17 2016: (Start)

Here are additional chains for the above list:

57120, 217728 with k = 13,14;

343976, 710400 with k = 7,8;

1980342, 5621760 with k = 10,14;

4404480, 14913024 with k = 11,12;

238608384, 775898880 with k = 11,12. (End)

Currently, the coefficient pairs are [1, 1], [3, 4], [4, 7], [7, 8], [8, 9], [9, 10], [9, 13], [10, 14], [11, 12], [13, 14], [16, 17], [16, 21], [17, 18], [19, 20], [23, 24], [25, 26], [25, 31], [27, 28], [29, 30], [31, 32], [32, 33], [37, 38]. It is interesting to note that for some of them, the pair (s,t) also satisfies t=sigma(s). - Michel Marcus, Jul 03 2016; Sep 06 2016

Using these empirical pairs of coefficients in conjunction with the first comment allows us to determine whether some term is the sum of divisors of another yet unknown smaller term. - Michel Marcus, Jul 04 2016

For m in A090748 = A000043 - 1 and c in A205597 (= odd a(n)), c*2^m is in the sequence, unless 2^(m+1)-1 | sigma(c). Indeed, from sigma(x*y) = sigma(x)*sigma(y) for gcd(x,y) = 1, we get sigma(sigma(c*2^m)) = sigma(sigma(c))*2^(m+1), so c*2^m is in the sequence if sigma(sigma(c))/c = k/2 (where k can't be odd: A330598 has no odd c). - M. F. Hasler, Jan 06 2020

It is conjectured that there are only finitely many terms. - N. J. A. Sloane, Jul 22 2012

Odd perfect number (unlikely to exist) and infinitely many Mersenne primes will make the sequence infinite - take the product of the OPN and coprime EPNs.

Conjecture: A010888(a(n)) divides a(n). Tested for n up to 36 incl. - Ivan N. Ianakiev, Oct 31 2013

From Farideh Firoozbakht, Dec 26 2014: (Start)

Theorem: If k>1 and p=a(n)/2^(k-2)+1 is prime then for each positive integer m, 2^(k-1)*p^m is a solution to the equation sigma(phi(x))=2*x-2^k, which implies the equation has infinitely many solutions.

Proof: sigma(phi(2^(k-1)*p^m)) = sigma(2^(k-2)*(p-1)*p^(m-1)) = sigma(2^(k-2)*(p-1))*sigma(p^(m-1)) = sigma(a(n))*(p^m-1)/(p-1) = 4*a(n)*(p^m-1)/(p-1) = 2^k*(p^m-1) = 2*(2^(k-1)*p^m)-2^k.

It seems that for all such equations there exist such an infinite set of solutions. So I conjecture that the sequence is infinite! (End)

If 3 were prepended to this sequence, then it would be the sequence of integers k such that numerator(sigma(k)/k)=4. - Michel Marcus, Nov 22 2015

Conjectured finite and probably these are the only terms; cf. Flammenkamp's link. [Georgi Guninski, Jul 25 2012]

Davenport shows that these numbers have positive density. Are there good bounds for the density?

G. Miller & M. Whalen suggested that 1018976683725 (3^3*5^2*7^2*11*13*17*19*23*29) might be the smallest odd number in the sequence (a fact now, see A119240 and A023197). - Michel Marcus, May 01 2013

From Amiram Eldar, Feb 13 2021: (Start)

Behrend (1933) found the bounds (0.009, 0.110) for the asymptotic density.

Wall et al. (1972) found the bounds (0.0186, 0.0461).

The upper bound was reduced to 0.0214614 using Deléglise's method by McDaniel College (2010). (End)

Note that 1018976683725, the smallest odd term in this sequence, is A053624(51). - Charles R Greathouse IV, Jan 09 2025

Named near-perfect numbers by sequence author.

Union of this sequence and A005820 is A153501.

Every even perfect number n = 2^(p-1)*(2^p-1), p and 2^p-1 prime, of A000396 generates three entries: 2*n, 2^p*n and (2^p-1)*n.

Every number M=2^(t-1)*P, where P is a prime of the form 2^t-2^k-1, is an entry for which (2^k)|M and sigma(M)-2^k=2*M (see A181701).

Conjecture 1: For every k>=1, there exist infinitely many entries m for which (2^k)|m and sigma(m)-2^k = 2*m.

Conjecture 2. All entries are even. [Proved to be false, see below. (Ed.)]

Conjecture 3. If the suitable (according to the definition) divisor d of an entry is not a power of 2, then it is not suitable divisor for any other entry.

Conjecture 4. If a suitable divisor for an even entry is odd, then it is a Mersenne prime (A000043).

If Conjectures 3 and 4 are true, then an entry with odd suitable divisor has the form 2^(p-1)*(2^p-1)^2, where p and 2^p-1 are primes. - Vladimir Shevelev, Nov 08 2010 to Dec 16 2010

The only odd term in this sequence < 2*10^12 is 173369889. - Donovan Johnson, Feb 15 2012

173369889 remains only odd term up to 1.4*10^19. - Peter J. C. Moses, Mar 05 2012

These numbers are obviously pseudoperfect (A005835) since they are equal to the sum of all the proper divisors except the one that is the same as the abundance. - Alonso del Arte, Jul 16 2012

On the Riemann Hypothesis, a(n) > exp(exp(n / e^gamma)) for n > 3. Unconditionally, a(n) > exp(exp(0.9976n / e^gamma)) for n > 3, where the constant is related to A004394(1000000). - Charles R Greathouse IV, Sep 06 2012

Each of the terms 1, 6, 120, 30240 divides all larger terms <= a(8). See A227765, A227766, ..., A227769. - Jonathan Sondow, Jul 30 2013

Is a(n) < a(n+1)? - Jeppe Stig Nielsen, Jun 16 2015

Equivalently, a(n) is the smallest number k such that sigma(k)/k = n. - Derek Orr, Jun 19 2015

The number of divisors of these terms are: 1, 4, 16, 96, 1920, 110592, 1751777280, 63121588161085440. - Michel Marcus, Jun 20 2015

Given n, let S_n be the sequence of integers k that satisfy numerator(sigma(k)/k) = n. Then a(n) is a member of S_n. In fact a(n) = S_n(i), where the successive values of i are 1, 1, 2, 2, 4, 2, (23, 6, 31, 12, ...), where the terms in parentheses need to be confirmed. - Michel Marcus, Nov 22 2015

The first four terms are the only multiperfect numbers in A025487 among the 1600 initial terms of A007691. Can it be proved that these are the only ones among the whole A007691? See also A349747. - Antti Karttunen, Dec 04 2021