A007539 a(n) = first n-fold perfect (or n-multiperfect) number.

1, 6, 120, 30240, 14182439040, 154345556085770649600, 141310897947438348259849402738485523264343544818565120000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• more

Comments

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

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
• A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 138.
• R. K. Guy, Unsolved Problems in Number Theory, B2.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..8
• Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
• Jamie Bishop, Abigail Bozarth, Rebekah Kuss, and Benjamin Peet, The Abundancy Index and Feebly Amicable Numbers, arXiv:2104.11366 [math.NT], 2021.
• C. K. Caldwell, The Prime Glossary, multiply perfect
• F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
• Achim Flammenkamp, The Multiply Perfect Numbers Page
• Fred Helenius, Link to Glossary and Lists
• G. P. Michon, Multiperfect and hemiperfect numbers
• Walter Nissen, Abundancy: Some Resources

Crossrefs

Cf. A000396, A005820, A007691, A025487, A027687, A046060, A046061, A072002, A227765, A227766, A227767, A227768, A227769, A349747.

Programs

Mathematica

Table[k = 1; While[DivisorSigma[1, k]/k != n, k++]; k, {n, 4}] (* Michael De Vlieger, Jun 20 2015 *)

PARI

a(n)=k=1;while((sigma(k)/k)!=n,k++);k
vector(4,n,a(n)) \\ Derek Orr, Jun 19 2015

Extensions

More terms sent by Robert G. Wilson v, Nov 30 2000