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%I N0213 #56 Sep 05 2020 20:02:08
%S 1,2,3,4,8,10,12,19,37,54,65,77,314,366,770,1327,1373,1937,2561,2663,
%T 5834,5985,6751,12003,13066,13973,26790,51924,66530,79502,130100,
%U 455663,517430,757263,841842,1791864,1819050,4197919,8107892,12640858,14471465,15632458,18304103,19616714,22370543
%N Number of digits in n-th even perfect number (A000396).
%C The next known values following a(47) are 34850340, 44677235 and 46498850, but these may not be the next terms. [Updated by _M. F. Hasler_, Nov 28 2017, _Ivan Panchenko_, Apr 07 2018, Apr 17 2018]
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.
%D Martin Gardner, Mathematical Magic Show. Alfred A. Knopf. 1977 p. 165
%D Paul Hoffman, Archimedes' Revenge. Penguin. 1988 p. 11
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D Donald D. Spencer, Key Dates in Number Theory History. Camelot Pub.Co. 1995 p. 80.
%H Ivan Panchenko, <a href="/A061193/b061193.txt">Table of n, a(n) for n = 1..47</a> [Updated by _Ivan Panchenko_, Apr 17 2018]
%H A. Byerley, <a href="https://web.archive.org/web/20010803230955/http://www.andrews.edu:80/~byerley/perfect.html">Relationship Between Mersenne Primes and Perfect Numbers</a>
%H C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html#known">Table of Known Mersenne Primes</a>.
%H J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> [Broken link]
%H J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> [Via Internet Archive Wayback-Machine]
%H J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a> [Cached copy, pdf file only]
%H H. J. Smith, <a href="http://harry-j-smith-memorial.com/Perfect/Mersenne.html">Mersenne Primes</a>
%F a(n) = ceiling((2*A000043(n)-1)*A007524), with A000043 = Mersenne prime exponents, A007524 = log_10(2). - _M. F. Hasler_, Nov 28 2017
%t Table[n=MersennePrimeExponent@k;IntegerLength[2^(n-1)(2^n-1)],{k,45}] (* _Giorgos Kalogeropoulos_, Sep 03 2020 *)
%t Array[IntegerLength@*PerfectNumber, 18] (* _Giorgos Kalogeropoulos_, Sep 03 2020 *)
%o (PARI) apply( p->(2*p-1)*log(2)\log(10)+1, A000043) \\ where A000043 is the vector of the known Mersenne primes. - _M. F. Hasler_, Nov 28 2017
%Y Cf. A000396.
%K nonn,base
%O 1,2
%A _N. J. A. Sloane_. This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by _Lekraj Beedassy_, May 30 2001.
%E More terms from _Harry J. Smith_, Apr 16 2003
%E Entry revised by _N. J. A. Sloane_, Jun 10 2012
%E a(39) through a(45) from _M. F. Hasler_, Nov 28 2017
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