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A081357
Sublime numbers, numbers for which the number of divisors and the sum of the divisors are both perfect.
11
12, 6086555670238378989670371734243169622657830773351885970528324860512791691264
OFFSET
1,1
COMMENTS
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
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
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
REFERENCES
David J. Darling, The universal book of mathematics: from Abracadabra to Zeno's paradoxes, Hoboken, N.J.: Wiley, 2004, p. 307.
Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, pp. 4 and 395.
Roozbeh Hazrat, Mathematica®: A Problem-Centered Approach, Springer, 2016, exercise 5.5, p. 102.
Clifford A. Pickover, Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning, Oxford University Press, 2001, p. 215.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 22.
Simon Singh, The Simpsons and Their Mathematical Secrets, A&C Black, 2013, p. 98.
LINKS
Kevin S. Brown, Twelve is special, posting to sci.math newsgroup, Mar 20 1995.
Kevin S. Brown, Odd Sublime Numbers?, posting to sci.math newsgroup, Mar 26 1995.
Kevin S. Brown, Sublime Numbers.
Jonny Griffiths, Lopsided numbers, Mathematical Spectrum, Vol. 43, No. 2 (2010/2011), pp. 53-54; entire issue.
Michael Joseph Halm, More Sequences, Mpossibilities, Issue 83, April 2003.
Dean Hickerson, Re: Twelve is special, posting to sci.math newsgroup, Mar 23 1995.
Pallavi Pathak and Jawahar Pathak, An algorithm to construct Sublime Numbers, Mathematics Today, Vol. 32 (2016), pp. 41-46.
Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
Gérard Villemin, Nombre sublime. (French)
Eric Weisstein's World of Mathematics, Sublime Number.
Wikipedia, Sublime number.
EXAMPLE
a(1) = 12 because 12 + 6 + 4 + 3 + 2 + 1 = 28 is perfect and number of divisors, 6, is also perfect.
CROSSREFS
KEYWORD
hard,nonn,bref,more
AUTHOR
Michael Joseph Halm, Apr 20 2003
STATUS
approved