Search: id:A000396 Results 1-1 of 1 results found. %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, Table of n, a(n) for n = 1..14 %H A000396 Walter Nissen, Abundancy : Some Resources %H A000396 Anonymous, Perfect Numbers %H A000396 Anonymous, Timetable of discovery of perfect numbers %H A000396 R. P. Brent & G. L. Cohen, A new lower bound for odd perfect numbers %H A000396 R. P. Brent, G. L. Cohen & H. J. J. te Riele, A new approach to lower bounds for odd perfect numbers %H A000396 R. P. Brent, G. L. Cohen & H. J. J. te Riele, Improved Techniques For Lower Bounds For Odd Perfect Numbers %H A000396 J. Britton, Perfect Number Analyser %H A000396 C. K. Caldwell, Perfect number %H A000396 C. K. Caldwell, Mersenne Primes, etc. %H A000396 C. K. Caldwell, Iterated sums of the digits of a perfect number converge to 1 %H A000396 S. Davis, A Rationality Condition for the Existence of Odd Perfect Numbers %H A000396 S. Davis, A Proof of the Odd Perfect Number Conjecture %H A000396 J. W. Gaberdiel, A Study of Perfect Numbers and Related Topics %H A000396 T. Goto & Y. Ohno, Largest prime factor of an odd perfect number %H A000396 K. G. Hare, New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number %H A000396 D. & C. Hazzlewood, Perfect Numbers [Broken link] %H A000396 D. & C. Hazzlewood, Perfect Numbers [Cached copy] %H A000396 C.-E. Jean, "Recreomath" Online Dictionary, Nombre parfait %H A000396 T. Leinster, Perfect numbers and groups. %H A000396 T. Masiwa, T. Shonhiwa & G. Hitchcock, Perfect Numbers & Mersenne Primes %H A000396 Mathforum, Perfect Numbers %H A000396 Mathforum, List of Perfect Numbers %H A000396 J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3 %H A000396 G. P. Michon, Topic 16:Perfect Numbers, Mersenne Primes %H A000396 D. Moews, Perfect, amicable and sociable numbers %H A000396 P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors %H A000396 J. J. O'Connor & E. F. Robertson, Perfect Numbers %H A000396 H. Ok, The Perfect Number Journey %H A000396 J. O. M. Pedersen, Perfect numbers %H A000396 J. O. M. Pedersen, Tables of Aliquot Cycles %H A000396 I. Peterson, Cubes of Perfection %H A000396 J. Perry, OddPerfect Numbers %H A000396 O. E. Pol, Determinacion geometrica de los numeros primos y perfectos. %H A000396 K. Schneider, PlanetMath.org, perfect number %H A000396 G. Villemin's Almanach of Numbers, Nombres Parfaits %H A000396 J. Voight, Perfect Numbers:An Elementary Introduction %H A000396 Eric Weisstein's World of Mathematics, Perfect Number %H A000396 Eric Weisstein's World of Mathematics, Odd Perfect Number %H A000396 Eric Weisstein's World of Mathematics, Multiperfect Number %H A000396 Eric Weisstein's World of Mathematics, Hyperperfect Number %H A000396 Eric Weisstein's World of Mathematics, Abundance %H A000396 Wikipedia, Perfect number %H A000396 T. Yamada, On the divisibility of odd perfect numbers by a high power of a prime %H A000396 Index entries for "core" sequences %H A000396 D. Romagnoli, Perfect Numbers (Text in Italian) [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 Search completed in 0.003 seconds