1,1

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 (blekraj(AT)yahoo.com), 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 (blekraj(AT)yahoo.com), Aug 21 2004

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

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

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

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

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]

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]

Contribution from Reikku Kulon (reikku(AT)gmail.com), 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. [From Omar E. Pol (info(AT)polprimos.com), Jan 14 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

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.

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

Walter Nissen, Abundancy : Some Resources

Anonymous, Perfect Numbers

Anonymous, Timetable of discovery of perfect numbers

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

S. Davis, A Rationality Condition for the Existence of Odd Perfect Numbers

S. Davis, A Proof of the Odd Perfect Number Conjecture

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

D. & C. Hazzlewood, Perfect Numbers [Broken link]

D. & C. Hazzlewood, Perfect Numbers [Cached copy]

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

T. Leinster, Perfect numbers and groups.

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, Topic 16:Perfect Numbers, Mersenne Primes

D. Moews, Perfect, amicable and sociable numbers

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

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

H. Ok, The Perfect Number Journey

J. O. M. Pedersen, Perfect numbers

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

I. Peterson, Cubes of Perfection

J. Perry, OddPerfect Numbers

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

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

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

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 (benoit7848c(AT)orange.fr), 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 (blekraj(AT)yahoo.com), Sep 21 2004

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

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

a(n) = A000217(A000668(n)), assuming there are no odd perfect numbers. - Omar E. Pol (info(AT)polprimos.com), May 09 2008

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

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]

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]

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]

a(n) = A007691(A153800(n)), assuming there are no odd perfect numbers. [From Omar E. Pol (info(AT)polprimos.com), 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). [From Lekraj Beedassy (blekraj(AT)yahoo.com), May 02 2009]

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.

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

(# (# + 1)/2 &/@ Select[FoldList[Plus, 0, NestList[2 # &, 1, 500]], PrimeQ] - Harvey P. Dale Mar 06 2002

Contribution from Michael Porter (michael_b_porter(AT)yahoo.com), Nov 03 2009: (Start)

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

forprime(p=1, 90, if(isprime(2^p-1), print(2^(p-1)*(2^p-1)))) (End)

See A000043 for the current state of knowledge about Mersenne primes. Cf. A007539, A005820, A027687, A046060, A046061.

Cf. A000668, A090748, A133033.

Cf. A000217.

Cf. A000384, A019279, A061652. [From Omar E. Pol (info(AT)polprimos.com), Aug 17 2008]

Cf. A006516. [From Omar E. Pol (info(AT)polprimos.com), Aug 30 2008]

Cf. A144912 [From Reikku Kulon (reikku(AT)gmail.com), Oct 14 2008]

Cf. A007691, A153800. [From Omar E. Pol (info(AT)polprimos.com), Jan 14 2009]

Sequence in context: A104511 A138876 A060286 this_sequence A152953 A066239 A097464

Adjacent sequences: A000393 A000394 A000395 this_sequence A000397 A000398 A000399

nonn,nice,core,new

N. J. A. Sloane (njas(AT)research.att.com).

I edited my comments and formulae - Omar E. Pol (info(AT)polprimos.com), Apr 22 2009, Apr 23 2009