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

12, 6086555670238378989670371734243169622657830773351885970528324860512791691264

Offset: 1

Author

Michael Joseph Halm, Apr 20 2003

Keywords

• hard
• nonn
• bref
• more

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

Examples

			a(1) = 12 because 12 + 6 + 4 + 3 + 2 + 1 = 28 is perfect and number of divisors, 6, is also perfect.
		

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.

Crossrefs

Cf. A000005, A000203, A000396.