cp's OEIS Frontend

A081357 Sublime numbers, numbers for which the number of divisors and the sum of the divisors are both perfect.

%I A081357 #38 Mar 06 2025 11:04:19
%S A081357 12,
%T A081357 6086555670238378989670371734243169622657830773351885970528324860512791691264
%N A081357 Sublime numbers, numbers for which the number of divisors and the sum of the divisors are both perfect.
%C A081357 The concept was introduced and the term "sublime numbers" was coined by Kevin Brown. a(1) was found by Brown (1995) and a(2) by Hickerson (1995). - _Amiram Eldar_, Jun 26 2021
%C A081357 a(2) = 2^126 * M(3) * M(5) * M(7) * M(19) * M(31) * M(61), where M(n) = 2^n - 1. - _Lucas A. Brown_, Mar 05 2025
%C A081357 It is known that all sublime numbers must be of that form, with all odd prime factors being Mersenne primes M(p). - _M. F. Hasler_, Mar 05 2025
%D A081357 David J. Darling, The universal book of mathematics: from Abracadabra to Zeno's paradoxes, Hoboken, N.J.: Wiley, 2004, p. 307.
%D A081357 Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, pp. 4 and 395.
%D A081357 Roozbeh Hazrat, Mathematica®: A Problem-Centered Approach, Springer, 2016, exercise 5.5, p. 102.
%D A081357 Clifford A. Pickover, Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning, Oxford University Press, 2001, p. 215.
%D A081357 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 22.
%D A081357 Simon Singh, The Simpsons and Their Mathematical Secrets, A&C Black, 2013, p. 98.
%H A081357 Kevin S. Brown, <a href="https://groups.google.com/g/sci.math/c/i_Kk9n0Nsl4/m/5dI7uEpXuEUJ">Twelve is special</a>, posting to sci.math newsgroup, Mar 20 1995.
%H A081357 Kevin S. Brown, <a href="https://groups.google.com/g/sci.math/c/TvXa5IdKf4Q/m/i6hvWgxw5UsJ">Odd Sublime Numbers?</a>, posting to sci.math newsgroup, Mar 26 1995.
%H A081357 Kevin S. Brown, <a href="http://www.mathpages.com/home/kmath202/kmath202.htm">Sublime Numbers</a>.
%H A081357 Jonny Griffiths, Lopsided numbers, Mathematical Spectrum, Vol. 43, No. 2 (2010/2011), pp. 53-54; <a href="http://www.appliedprobability.org/data/files/MS%20issues/Vol43_No2.pdf">entire issue</a>.
%H A081357 Michael Joseph Halm, <a href="https://michaelhalm.tripod.com/id171.htm">More Sequences</a>, Mpossibilities, Issue 83, April 2003.
%H A081357 Dean Hickerson, <a href="https://groups.google.com/g/sci.math/c/i_Kk9n0Nsl4/m/UHJTt-P7ePUJ">Re: Twelve is special</a>, posting to sci.math newsgroup, Mar 23 1995.
%H A081357 Pallavi Pathak and Jawahar Pathak, <a href="https://web.archive.org/web/20200714113410/http://mathematicstoday.org/archives/V32_2016_5.pdf">An algorithm to construct Sublime Numbers</a>, Mathematics Today, Vol. 32 (2016), pp. 41-46.
%H A081357 Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0983.00008">Zentralblatt review</a>.
%H A081357 Gérard Villemin, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Decompos/Sublime.htm">Nombre sublime</a>. (French)
%H A081357 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SublimeNumber.html">Sublime Number</a>.
%H A081357 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sublime_number">Sublime number</a>.
%e A081357 a(1) = 12 because 12 + 6 + 4 + 3 + 2 + 1 = 28 is perfect and number of divisors, 6, is also perfect.
%Y A081357 Cf. A000005, A000203, A000396.
%K A081357 hard,nonn,bref,more
%O A081357 1,1
%A A081357 _Michael Joseph Halm_, Apr 20 2003