A227766 -id:A227766 - cp's OEIS frontend

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

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

Offset: 1

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

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).

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

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

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

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

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

A227765 First n-fold perfect number divided by 6, or 0 if the quotient is not an integer.

Original entry on oeis.org

0, 1, 20, 5040, 2363739840, 25724259347628441600, 23551816324573058043308233789747587210723924136427520000

Offset: 1

Jonathan Sondow, Jul 29 2013

nonn

The first n-fold perfect numbers are A007539(n) = 1, 6, 120, 30240, 14182439040, ..., for n = 1, 2, 3, 4, 5, ...
The first n-fold perfect numbers divided by 120 is A227766.
The first n-fold perfect numbers divided by 30240 is A227767.
The first n-fold perfect numbers divided by 14182439040 is A227768.
For additional comments, references, links, and cross-refs, see A007539.

			The first 3-fold perfect number is A007539(3) = 120, so a(3) = 120/6 = 20.

T. D. Noe, Table of n, a(n) for n = 1..8
E. W. Weisstein's MathWorld, Multiperfect Number
Wikipedia, Multiply perfect number

Cf. A007539, A227766, A227767, A227768, A227769.

a(n) = A007539(n)/6 if 2 <= n <= 8.

A227767 First n-fold perfect number divided by 30240, or 0 if the quotient is not an integer.

Original entry on oeis.org

0, 0, 0, 1, 468996, 5104019711831040, 4672979429478781357799252736061029208476969074688000

Offset: 1

Jonathan Sondow, Jul 29 2013

nonn

The first n-fold perfect numbers are A007539(n) = 1, 6, 120, 30240, 14182439040, ..., for n = 1, 2, 3, 4, 5, ...

For additional comments, references, links, and cross-refs, see A007539 and A227765.

			The first 5-fold perfect number is A007539(5) = 14182439040, so a(5) = 14182439040/30240 = 468996.

T. D. Noe, Table of n, a(n) for n = 1..8

Cf. A007539, A227765, A227766, A227768, A227769.

a(n) = A007539(n)/30240 if 4 <= n <= 8.

A227768 First n-fold perfect number divided by 14182439040, or 0 if the quotient is not an integer.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 9963793783910270786529635084437882643939328000

Offset: 1

Jonathan Sondow, Jul 29 2013

nonn

The first n-fold perfect numbers are A007539(n) = 1, 6, 120, 30240, 14182439040, .., for n = 1, 2, 3, 4, 5, ...

For additional comments, references, links, and cross-refs, see A007539 and A227765.

			The first 6-fold perfect number is A007539(6) = 154345556085770649600, and 154345556085770649600/14182439040 = 119711504640/11 is not an integer, so a(6) = 0.

T. D. Noe, Table of n, a(n) for n = 1..8

Cf. A007539, A227765, A227766, A227767, A227769.

a(n) = A007539(n)/14182439040 for n = 5, 7, 8.

A227769 First (n+1)-fold perfect number divided by first n-fold perfect number, or 0 if the quotient is not an integer.

Original entry on oeis.org

6, 20, 252, 468996, 0, 0, 0

Offset: 1

Jonathan Sondow, Jul 30 2013

nonn

The first n-fold perfect numbers are A007539 = 1, 6, 120, 30240, 14182439040, 154345556085770649600, ...

For additional comments, references, links, and crossrefs, see A007539 and A227765.

			The first 4-fold and 5-fold perfect numbers are A007539(4) = 30240 and  A007539(5) = 14182439040, and 14182439040/30240 = 468996 is an integer, so a(4) = 468996.

Cf. A007539, A227765, A227766, A227767, A227768.

a(n) = A007539(n+1)/A007539(n) if n < 5.

cp's OEIS Frontend

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

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A227765 First n-fold perfect number divided by 6, or 0 if the quotient is not an integer.

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A227767 First n-fold perfect number divided by 30240, or 0 if the quotient is not an integer.

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A227768 First n-fold perfect number divided by 14182439040, or 0 if the quotient is not an integer.

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A227769 First (n+1)-fold perfect number divided by first n-fold perfect number, or 0 if the quotient is not an integer.

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