cp's OEIS Frontend

A061193 Number of digits in n-th even perfect number (A000396).

%I A061193 N0213 #71 Jul 09 2025 01:40:56
%S A061193 1,2,3,4,8,10,12,19,37,54,65,77,314,366,770,1327,1373,1937,2561,2663,
%T A061193 5834,5985,6751,12003,13066,13973,26790,51924,66530,79502,130100,
%U A061193 455663,517430,757263,841842,1791864,1819050,4197919,8107892,12640858,14471465,15632458,18304103,19616714,22370543
%N A061193 Number of digits in n-th even perfect number (A000396).
%C A061193 The next known values following a(48) are 44677235, 46498850, and 49724095, but these may not be the next terms. [Updated by _M. F. Hasler_, Nov 28 2017, _Ivan Panchenko_, Apr 07 2018, Apr 17 2018, _Amiram Eldar_, Oct 16 2024]
%D A061193 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.
%D A061193 Martin Gardner, Mathematical Magic Show, Alfred A. Knopf, 1977, p. 165.
%D A061193 Paul Hoffman, Archimedes' Revenge, Penguin, 1988, p. 11.
%D A061193 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 244-245.
%D A061193 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A061193 Donald D. Spencer, Key Dates in Number Theory History, Camelot Pub. Co., 1995, p. 80.
%H A061193 Ivan Panchenko, <a href="/A061193/b061193.txt">Table of n, a(n) for n = 1..48</a> [Updated by _Ivan Panchenko_, Apr 17 2018, _Amiram Eldar_, Oct 16 2024]
%H A061193 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 A061193 Chris K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html#known">Table of Known Mersenne Primes</a>.
%H A061193 J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Broken link]
%H A061193 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 A061193 J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a>. [Cached copy, pdf file only]
%H A061193 H. J. Smith, <a href="http://harry-j-smith-memorial.com/Perfect/Mersenne.html">Mersenne Primes</a>. [broken link]
%F A061193 a(n) = ceiling((2*A000043(n)-1)*A007524), with A000043 = Mersenne prime exponents, A007524 = log_10(2). - _M. F. Hasler_, Nov 28 2017
%t A061193 Table[n=MersennePrimeExponent@k;IntegerLength[2^(n-1)(2^n-1)],{k,45}] (* _Giorgos Kalogeropoulos_, Sep 03 2020 *)
%t A061193 Array[IntegerLength@*PerfectNumber, 18] (* _Giorgos Kalogeropoulos_, Sep 03 2020 *)
%o A061193 (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 A061193 Cf. A000043, A000396, A007524, A055642.
%K A061193 nonn,base
%O A061193 1,2
%A A061193 _N. J. A. Sloane_
%E A061193 This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by _Lekraj Beedassy_, May 30 2001
%E A061193 More terms from _Harry J. Smith_, Apr 16 2003
%E A061193 Entry revised by _N. J. A. Sloane_, Jun 10 2012
%E A061193 a(39) through a(45) from _M. F. Hasler_, Nov 28 2017