%I A000396 M4186 N1744
%S A000396 6,28,496,8128,33550336,8589869056,137438691328,
%T A000396 2305843008139952128,2658455991569831744654692615953842176,
%U A000396 191561942608236107294793378084303638130997321548169216
%N A000396 Perfect numbers n: n is equal to the sum of the proper divisors of n.
%C A000396 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).
%C A000396 For number of divisors of a(n) see A061645(n). Number of digits in a(n)
is A061193(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 04 2004
%C A000396 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 (blekraj(AT)yahoo.com), Aug 21 2004
%C A000396 The earliest recorded mention of this sequence is in Euclid's Elements,
IX 36, about 300 BC. - Artur Jasinski (grafix(AT)csl.pl), Jan 25
2006
%C A000396 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 (info(AT)polprimos.com), Feb 28 2008
%C A000396 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 (azarian(AT)evansville.edu),
Apr 16 2008
%C A000396 Triangular numbers A000217 whose indices are Mersenne primes A000668,
assuming there are no odd perfect numbers. 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 (info(AT)polprimos.com),
May 09 2008
%C A000396 Hexagonal numbers A000384 whose indices are superperfect numbers A019279,
assuming there are no odd perfect numbers and no odd superperfect
numbers. [From Omar E. Pol (info(AT)polprimos.com), Aug 17 2008]
%C A000396 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. [From Omar E. Pol (info(AT)polprimos.com), Aug 30 2008]
%C A000396 Contribution from Reikku Kulon (reikku(AT)gmail.com), Oct 14 2008: (Start)
%C A000396 A144912(2, a(n)) = 1;
%C A000396 A144912(4, a(n)) = -1 for n > 1;
%C A000396 A144912(8, a(n)) = 5 or -5 for all n except 2;
%C A000396 A144912(16, a(n)) = -4 or -13 for n > 1. (End)
%C A000396 Multiply-perfect numbers A007691 whose indices are the numbers A153800,
assuming there are no odd perfect numbers. [From Omar E. Pol (info(AT)polprimos.com),
Jan 14 2009]
%D A000396 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000396 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000396 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 4.
%D A000396 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964,
p. 19.
%D A000396 S. Bezuszka, Perfect Numbers, (Booklet 3, Motivated Math. Project Activities)
Boston College Press, Chestnut Hill MA 1980.
%D A000396 Euclid, Elements, Book IX, Section 36, about 300 BC.
%D A000396 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers.
3rd ed., Oxford Univ. Press, 1954, p. 239.
%D A000396 T. Koshy, "The Ends Of A Mersenne Prime And An Even Perfect Number",
Journal of Recreational Mathematics, pp. 196-202 Baywood NY 1998.
%D A000396 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)
%D A000396 J. Sandor, Handbook of Number Theory, II, Springer Verlag, 2004.
%D A000396 I. Stewart, L'univers des nombres, "Diviser Pour Regner", Chapter 14,
pp. 74-81 Belin-Pour La Science, Paris 2000.
%D A000396 H. S. Uhler, On the 16th and 17th perfect numbers, Scripta Math. 19 (1953),
128-131.
%D A000396 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers,
pp. 107-110 Penguin Books 1987.
%H A000396 David Wasserman, <a href="b000396.txt">Table of n, a(n) for n = 1..14</
a>
%H A000396 Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">
Abundancy : Some Resources </a>
%H A000396 Anonymous, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/node3.html">
Perfect Numbers</a>
%H A000396 Anonymous, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/node43.html">
Timetable of discovery of perfect numbers</a>
%H A000396 R. P. Brent & G. L. Cohen, <a href="http://wwwmaths.anu.edu.au/~brent/
pub/pub100.html">A new lower bound for odd perfect numbers</a>
%H A000396 R. P. Brent, G. L. Cohen & H. J. J. te Riele, <a href="http://wwwmaths.anu.edu.au/
~brent/pub/pub106.html">A new approach to lower bounds for odd perfect
numbers</a>
%H A000396 R. P. Brent, G. L. Cohen & H. J. J. te Riele, <a href="http://db.cwi.nl/
rapporten/abstract.php?abstractnr=1802">Improved Techniques For Lower
Bounds For Odd Perfect Numbers</a>
%H A000396 J. Britton, <a href="http://britton.disted.camosun.bc.ca/perfect/jbperfect.htm">
Perfect Number Analyser</a>
%H A000396 C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=PerfectNumber">
Perfect number</a>
%H A000396 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/
index.html">Mersenne Primes, etc.</a>
%H A000396 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/notes/proofs/
Theorem3.html">Iterated sums of the digits of a perfect number converge
to 1</a>
%H A000396 S. Davis, <a href="http://xxx.lanl.gov/abs/math.NT/0011206">A Rationality
Condition for the Existence of Odd Perfect Numbers</a>
%H A000396 S. Davis, <a href="http://arXiv.org/abs/hep-th/0401052">A Proof of the
Odd Perfect Number Conjecture</a>
%H A000396 J. W. Gaberdiel, <a href="http://math.arizona.edu/~ura/001/gaberdiel.jw/
">A Study of Perfect Numbers and Related Topics</a>
%H A000396 T. Goto & Y. Ohno, <a href="http://www.ma.noda.tus.ac.jp/u/tg/perfect.html">
Largest prime factor of an odd perfect number</a>
%H A000396 K. G. Hare, <a href="http://fr.arXiv.org/abs/math.NT/0501070">New techniques
for bounds on the total number of Prime Factors of an Odd Perfect
Number</a>
%H A000396 D. & C. Hazzlewood, <a href="http://www.math.swt.edu/~haz/prob_sets/notes/
node13.html">Perfect Numbers</a> [Broken link]
%H A000396 D. & C. Hazzlewood, <a href="a000396_h.html">Perfect Numbers</a> [Cached
copy]
%H A000396 C.-E. Jean, "Recreomath" Online Dictionary, <a href="http://www.recreomath.qc.ca/
dict_parfait_nombre.htm">Nombre parfait</a>
%H A000396 T. Leinster, <a href="http://arXiv.org/abs/math.GR/0104012">Perfect numbers
and groups</a>.
%H A000396 T. Masiwa, T. Shonhiwa & G. Hitchcock, <a href="http://uzweb.uz.ac.zw/
science/maths/zimaths/51/perfect.htm">Perfect Numbers & Mersenne
Primes</a>
%H A000396 Mathforum, <a href="http://mathforum.org/dr.math/faq/faq.perfect.html">
Perfect Numbers</a>
%H A000396 Mathforum, <a href="http://mathforum.org/library/drmath/view/51516.html">
List of Perfect Numbers</a>
%H A000396 J. S. McCranie, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3</
a>
%H A000396 G. P. Michon, <a href="http://home.att.net/~numericana/answer/numbers.htm">
Topic 16:Perfect Numbers, Mersenne Primes</a>
%H A000396 D. Moews, <a href="http://djm.cc/amicable.html">Perfect, amicable and
sociable numbers</a>
%H A000396 P. P. Nielsen, <a href="http://fr.arXiv.org/abs/math.NT/0602485">Odd
Perfect Numbers Have At Least Nine Distinct Prime Factors</a>
%H A000396 J. J. O'Connor & E. F. Robertson, <a href="http://www-groups.dcs.st-and.ac.uk/
~history/HistTopics/Perfect_numbers.html">Perfect Numbers</a>
%H A000396 H. Ok, <a href="http://home1.pacific.net.sg/~novelway/MEW2/lesson1.html">
The Perfect Number Journey</a>
%H A000396 J. O. M. Pedersen, <a href="http://amicable.homepage.dk/perfect.htm">
Perfect numbers</a>
%H A000396 J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables
of Aliquot Cycles</a>
%H A000396 I. Peterson, <a href="http://www.maa.org/mathland/mathtrek_5_18_98.html">
Cubes of Perfection</a>
%H A000396 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/oddperfectnumbers/
oddperfectnumbers.htm">OddPerfect Numbers</a>
%H A000396 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica
de los numeros primos y perfectos</a>.
%H A000396 K. Schneider, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
PerfectNumber.html">perfect number</a>
%H A000396 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/
Decompos/SixNbPf.htm">Nombres Parfaits</a>
%H A000396 J. Voight, <a href="http://magma.maths.usyd.edu.au/~voight/notes/perfelem.pdf">
Perfect Numbers:An Elementary Introduction</a>
%H A000396 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PerfectNumber.html">Perfect Number</a>
%H A000396 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
OddPerfectNumber.html">Odd Perfect Number</a>
%H A000396 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
MultiperfectNumber.html">Multiperfect Number</a>
%H A000396 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
HyperperfectNumber.html">Hyperperfect Number</a>
%H A000396 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Abundance.html">Abundance</a>
%H A000396 Wikipedia, <a href="http://www.wikipedia.org/wiki/Perfect_number">Perfect
number</a>
%H A000396 T. Yamada, <a href="http://arXiv.org/abs/math.NT/0511410">On the divisibility
of odd perfect numbers by a high power of a prime</a>
%H A000396 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000396 D. Romagnoli, <a href="http://www.mistretta.eu/PDF/I%20numeri%20perfetti.pdf">
Perfect Numbers (Text in Italian)</a> [From Lekraj Beedassy (blekraj(AT)yahoo.com),
Jun 26 2009]
%F A000396 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.
%F A000396 Numbers n such that sum(d|n, 1/d)=2 - Benoit Cloitre (benoit7848c(AT)orange.fr),
Apr 07 2002
%F A000396 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 (blekraj(AT)yahoo.com),
Sep 21 2004
%F A000396 a(n) = 2^A133033(n) - 2^A090748(n), assuming there are no odd perfect
numbers. - Omar E. Pol (info(AT)polprimos.com), Feb 28 2008
%F A000396 a(n) = A000668(n)*(A000668(n)+1)/2, assuming there are no odd perfect
numbers. - Omar E. Pol (info(AT)polprimos.com), Apr 23 2008
%F A000396 a(n) = A000217(A000668(n)), assuming there are no odd perfect numbers.
- Omar E. Pol (info(AT)polprimos.com), May 09 2008
%F A000396 a(n) = Sum of first A000668(n) positive integers, assuming there are
no odd perfect numbers. - Omar E. Pol (info(AT)polprimos.com), May
09 2008
%F A000396 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. [From Omar E. Pol (info(AT)polprimos.com),
Aug 17 2008]
%F A000396 It appears that a(n) = A006516(A000043(n)), assuming there are no odd
perfect numbers. [From Omar E. Pol (info(AT)polprimos.com), Aug 30
2008]
%F A000396 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. [From Omar E. Pol (info(AT)polprimos.com),
Jan 09 2009]
%F A000396 a(n) = A007691(A153800(n)), assuming there are no odd perfect numbers.
[From Omar E. Pol (info(AT)polprimos.com), Jan 14 2009]
%F A000396 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).
[From Lekraj Beedassy (blekraj(AT)yahoo.com), May 02 2009]
%e A000396 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.
%p A000396 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
(zerinvarylajos(AT)yahoo.com), Feb 05 2008
%t A000396 (# (# + 1)/2 &/@ Select[FoldList[Plus, 0, NestList[2 # &, 1, 500]], PrimeQ]
- Harvey P. Dale Mar 06 2002
%o A000396 Contribution from Michael Porter (michael_b_porter(AT)yahoo.com), Nov
03 2009: (Start)
%o A000396 (PARI) isA000396(n) = (sigma(n) == 2*n)
%o A000396 forprime(p=1,90,if(isprime(2^p-1),print(2^(p-1)*(2^p-1)))) (End)
%Y A000396 See A000043 for the current state of knowledge about Mersenne primes.
Cf. A007539, A005820, A027687, A046060, A046061.
%Y A000396 Cf. A000668, A090748, A133033.
%Y A000396 Cf. A000217.
%Y A000396 Cf. A000384, A019279, A061652. [From Omar E. Pol (info(AT)polprimos.com),
Aug 17 2008]
%Y A000396 Cf. A006516. [From Omar E. Pol (info(AT)polprimos.com), Aug 30 2008]
%Y A000396 Cf. A144912 [From Reikku Kulon (reikku(AT)gmail.com), Oct 14 2008]
%Y A000396 Cf. A007691, A153800. [From Omar E. Pol (info(AT)polprimos.com), Jan
14 2009]
%Y A000396 Sequence in context: A104511 A138876 A060286 this_sequence A152953 A066239
A097464
%Y A000396 Adjacent sequences: A000393 A000394 A000395 this_sequence A000397 A000398
A000399
%K A000396 nonn,nice,core,new
%O A000396 1,1
%A A000396 N. J. A. Sloane (njas(AT)research.att.com).
%E A000396 I edited my comments and formulae - Omar E. Pol (info(AT)polprimos.com),
Apr 22 2009, Apr 23 2009
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