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A000396 Perfect numbers n: n is equal to the sum of the proper divisors of n.
(Formerly M4186 N1744)
311
6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A number n is abundant if sigma(n) > 2n (cf. A005101), perfect if sigma(n) = 2n (this entry), deficient if sigma(n) < 2n (cf. A005100), where sigma(n) is the sum of the divisors of n (A000203).

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 entries 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 Mersenne primes M=2^p - 1, for odd p, are of form 6*t+1. Thus perfect numbers N, being M-th triangular, have 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

The number of divisors of a(n) that are powers of 2 is equal to A000043(n), assuming there are no odd perfect numbers. The number of divisors of a(n) that are multiples of n-th Mersenne prime A000668(n) is also equal to A000043(n), again assuming there are no odd perfect numbers. - Omar E. Pol, Feb 28 2008

Theorem (Euler). An even number n is a perfect number if and only if n=2^(k-1)*(2^k-1), where 2^k-1 is prime. Euler's idea came from Euclid's Proposition 36 of Book IX. 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

Sum of first m positive integers, where m is the n-th Mersenne prime A000668(n), assuming there are no odd perfect numbers. - Omar E. Pol, May 09 2008

Hexagonal numbers A000384 whose indices are superperfect numbers A019279, assuming there are no odd perfect numbers and no odd superperfect numbers. [Omar E. Pol, Aug 17 2008]

It appears that this sequence is equal to the numbers A006516 whose indices are the prime numbers A000043, assuming there are no odd perfect numbers. [Omar E. Pol, Aug 30 2008]

Contribution 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)

Multiply-perfect numbers A007691 whose indices are the numbers A153800, assuming there are no odd perfect numbers. [Omar E. Pol, Jan 14 2009]

If a(n) is even, then 2*a(n) is in A181595. [Vladimir Shevelev, Nov 07 2010]

Except for 6, all even terms are of the form 30*k - 2 or 45*k + 1. [Arkadiusz Wesolowski, Mar 11 2012]

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.

S. Bezuszka, Perfect Numbers, (Booklet 3, Motivated Math. Project Activities) Boston College Press, Chestnut Hill MA 1980.

Euclid, Elements, Book IX, Section 36, about 300 BC.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 239.

T. Koshy, "The Ends Of A Mersenne Prime And An Even Perfect Number", Journal of Recreational Mathematics, pp. 196-202 Baywood NY 1998.

Joseph S. Madachy: Madachy's Mathematical Recreations. New York: Dover Publications, Inc., 1979, p. 149 (First publ. by Charles Scribner's Sons, New York, 1966, under the title: Mathematics on Vacation)

J. Sandor, Handbook of Number Theory, II, Springer Verlag, 2004.

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).

I. Stewart, L'univers des nombres, "Diviser Pour Regner", Chapter 14, pp. 74-81 Belin-Pour La Science, Paris 2000.

H. S. Uhler, On the 16th and 17th perfect numbers, Scripta Math. 19 (1953), 128-131.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 107-110 Penguin Books 1987.

LINKS

David Wasserman, Table of n, a(n) for n = 1..14

Anonymous, Perfect Numbers [broken link]

Anonymous, Timetable of discovery of perfect numbers [broken link]

R. P. Brent & G. L. Cohen, A new lower bound for odd perfect numbers

R. P. Brent, G. L. Cohen & H. J. J. te Riele, A new approach to lower bounds for odd perfect numbers

R. P. Brent, G. L. Cohen & H. J. J. te Riele, Improved Techniques For Lower Bounds For Odd Perfect Numbers

J. Britton, Perfect Number Analyser

C. K. Caldwell, Perfect number

C. K. Caldwell, Mersenne Primes, etc.

C. K. Caldwell, Iterated sums of the digits of a perfect number converge to 1

J. W. Gaberdiel, A Study of Perfect Numbers and Related Topics

T. Goto & Y. Ohno, Largest prime factor of an odd perfect number

K. G. Hare, New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number

C.-E. Jean, "Recreomath" Online Dictionary, Nombre parfait

T. Leinster, Perfect numbers and groups, arXiv:math.GR/0104012

T. Masiwa, T. Shonhiwa & G. Hitchcock, Perfect Numbers & Mersenne Primes

Mathforum, Perfect Numbers

Mathforum, List of Perfect Numbers

J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3

G. P. Michon, Perfect Numbers, Mersenne Primes

D. Moews, Perfect, amicable and sociable numbers

P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors

Walter Nissen, Abundancy : Some Resources

J. J. O'Connor & E. F. Robertson, Perfect Numbers

J. O. M. Pedersen, Perfect numbers

J. O. M. Pedersen, Tables of Aliquot Cycles

I. Peterson, Cubes of Perfection

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

D. Romagnoli, Perfect Numbers (Text in Italian) [From Lekraj Beedassy, Jun 26 2009]

D. Scheffler, R. Ondrejka, The numerical evaluation of the eighteenth perfect number, Math. Comp. 14 (70) (1960) 199-200

K. Schneider, PlanetMath.org, perfect number

G. Villemin's Almanach of Numbers, Nombres Parfaits

J. Voight, Perfect Numbers:An Elementary Introduction

Eric Weisstein's World of Mathematics, Perfect Number

Eric Weisstein's World of Mathematics, Odd Perfect Number

Eric Weisstein's World of Mathematics, Multiperfect Number

Eric Weisstein's World of Mathematics, Hyperperfect Number

Eric Weisstein's World of Mathematics, Abundance

Wikipedia, Perfect number

T. Yamada, On the divisibility of odd perfect numbers by a high power of a prime

Index entries for "core" sequences

FORMULA

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 n such that sum(d|n, 1/d)=2 - Benoit Cloitre, Apr 07 2002

The perfect number N={2^(p-1)}*(2^p - 1) is also multiplicatively p-perfect, (i.e. A007955(N)=N^p) since tau(N)=2p. - 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 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]

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 + 1) = (A060866(n + 1))/2. - Juri-Stepan Gerasimov, Dec 17 2013

EXAMPLE

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.

MAPLE

ZL:=[]: for p from 1 to 101 do if (isprime(p) and isprime(2^p-1)) then ZL:=[op(ZL), 2^(p-1)*(2^p-1)]; fi; od; print(ZL); # - Zerinvary Lajos, Feb 05 2008

MATHEMATICA

A000396 = (# (# + 1))/2 & /@ Select[2^Range[100] - 1, PrimeQ] (* Harvey P. Dale, Mar 06 2002, Apr 06 2011 *)

PROG

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

forprime(p=1, 90, if(isprime(2^p-1), print(2^(p-1)*(2^p-1)))) \\ Michael B. Porter, Nov 03 2009

(Haskell)

a000396 n = a000396_list !! (n-1)

a000396_list = [x | x <- [1..], a000203 x == 2 * x]

-- Reinhard Zumkeller, Jan 20 2012

CROSSREFS

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, A007691, A153800, A007593, A220290, A028499-A028502, A034916, A065549.

Sequence in context: A174633 A201186 A060286 * A152953 A066239 A097464

Adjacent sequences:  A000393 A000394 A000395 * A000397 A000398 A000399

KEYWORD

nonn,nice,core

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 28 00:26 EDT 2014. Contains 244987 sequences.