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

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

1,1

These are the non-deficient numbers.

Comment from Max Alekseyev (maxale(AT)gmail.com), Jan 26 2005: The sequence of n that give local minima for A004125, i.e. such that A004125(n-1)>A004125(n) and A004125(n)<A004125(n+1) coincides with this sequence for the first 1014 terms. Then there appears 4095 which is a term of A023196 but is not a local minima.

Also union of pseudoperfect and weird numbers. Cf. A005835, A006037. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 29 2006

T. D. Noe, Table of n, a(n) for n=1..1000

If n is a member so is every positive multiple of n. The "primitive" members form A006039.

Flatten[Table[If[DivisorSigma[1, n] >= 2*n, n, {}], {n, 1, 300}]] - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 18 2008

Cf. A004125, A006039, A000396, A005101.

Sequence in context: A051774 A119357 A097216 this_sequence A005835 A007620 A100715

Adjacent sequences: A023193 A023194 A023195 this_sequence A023197 A023198 A023199

nonn,nice

David W. Wilson (davidwwilson(AT)comcast.net)

1,1

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

a(n) = A000396(n+1) - A000396(n).

a(1)=22 because A000396(1)=6 and A000396(2)=28 then 28-6 = 22.

Cf. A000396, A139229, A139230, A139231, A139232, A139233, A139234, A139235, A139236, A139237.

Sequence in context: A170655 A170703 A170741 this_sequence A158535 A171327 A077421

Adjacent sequences: A139225 A139226 A139227 this_sequence A139229 A139230 A139231

nonn

Omar E. Pol (info(AT)polprimos.com), Apr 18 2008

1,1

a(n) = A139228(n)/2.

Cf. A000396, A139228, A139230, A139231, A139232, A139233, A139234, A139235, A139236, A139237.

Sequence in context: A033864 A142120 A091805 this_sequence A142672 A158297 A090921

Adjacent sequences: A139226 A139227 A139228 this_sequence A139230 A139231 A139232

nonn

Omar E. Pol (info(AT)polprimos.com), Apr 19 2008

1,1

Cf. A000396, A139228, A139229, A139231, A139232, A139233, A139234, A139235, A139236, A139237.

Sequence in context: A160583 A025354 A025346 this_sequence A065576 A147628 A153486

Adjacent sequences: A139227 A139228 A139229 this_sequence A139231 A139232 A139233

nonn

Omar E. Pol (info(AT)polprimos.com), Apr 19 2008

1,1

a(n) = A139230(n)/2.

Cf. A000396, A139228, A139229, A139230, A139231, A139232, A139234, A139235, A139236, A139237.

Sequence in context: A108819 A158226 A152834 this_sequence A153165 A094491 A162604

Adjacent sequences: A139230 A139231 A139232 this_sequence A139234 A139235 A139236

nonn

Omar E. Pol (info(AT)polprimos.com), Apr 19 2008

1,1

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.

p={ the list of the Mersenne exponents (A000043) }; Mod[(PowerMod[2, p, 1000] - 1)(PowerMod[2, p - 1, 1000]), 1000]

Cf. A094540, A000396. See A000043 for the present state of knowledge about Mersenne primes.

Sequence in context: A035527 A085844 A083387 this_sequence A138876 A060286 A000396

Adjacent sequences: A104508 A104509 A104510 this_sequence A104512 A104513 A104514

nonn,base

Alfred S. Posamentier (asp2(AT)juno.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 23 2005

1,1

Also, concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th superperfect number A019279(n) and final digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.

Also, concatenation of n-th term of A080172, A138125(n) and A094540(n).

a(1)=326. For n>1 a(n) is equal to 166 or 748, only.

L. C. Noll, Mersenne Prime Digits and Names.

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

Cf. A000396, A000668, A019279, A061652, A080172, A094540, A138125, A138817.

Sequence in context: A031516 A066128 A138816 this_sequence A158271 A097737 A126311

Adjacent sequences: A138814 A138815 A138816 this_sequence A138818 A138819 A138820

base,nonn

Omar E. Pol (info(AT)polprimos.com), Apr 01 2008

1,1

Conjecture: Ratio of 6's to 8's approaches 1.5 as a limit as this sequence approaches infinity - J. Lowell, jhbubby(AT)mindspring.com, Jun 26 2007

Math Forum at Drexel, List of Perfect Numbers

Eric Weisstein's World of Mathematics, Perfect Number

(Maple code from N. J. A. Sloane (njas(AT)research.att.com)) # let s1 := list of terms in A000043

f:=n->if n mod 4 = 0 then 4 else n mod 4; fi; g:= x->2^f(x-1)*(2^f(x)-1); map(g, s1);

p={ the list of the Mersenne exponents (A000043) }; Mod[(PowerMod[2, p, 10] - 1)(PowerMod[2, p - 1, 10]), 10] (from Robert G. Wilson v May 23 2004)

Cf. A000396. See A000043 for the present state of knowledge about Mersenne primes.

Sequence in context: A021597 A088751 A153627 this_sequence A010724 A065356 A019797

Adjacent sequences: A094537 A094538 A094539 this_sequence A094541 A094542 A094543

nonn,hard,base

Eric Weisstein (eric(AT)weisstein.com), May 08, 2004

Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 01 2008, at the suggestion of Ant King and Omar Pol.

0,1

The next term is too large to include. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 12 2006

Sequence in context: A084108 A006713 A006712 this_sequence A123278 A127605 A013975

Adjacent sequences: A099054 A099055 A099056 this_sequence A099058 A099059 A099060

nonn

N. J. A. Sloane (njas(AT)research.att.com), Nov 15 2004

More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 12 2006