Sublime number

An elevated number or sublime number is a natural number with the special property that the number of their division and its splitters sum perfect numbers are. So far only two raised numbers are known. ${\ displaystyle n}$ ${\ displaystyle \ tau (n)}$ ${\ displaystyle \ sigma (n)}$

Well-known raised numbers

12

The 12 has 6 factors: 12, 6, 4, 3, 2, 1.
The sum of their divisors is: 1 + 2 + 3 + 4 + 6 + 12 = 28.

{\ displaystyle \ tau (12) = 6}

and are perfect numbers.

{\ displaystyle \ sigma (12) = 28}

6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264

6086555670238378989670371734243169622657830773351885970528324860512791691264= 2 ¹²⁶ (2 ⁶¹ - 1) (2 ³¹ - 1) (2 ¹⁹ - 1) (2 ⁷ - 1) (2 ⁵ - 1) (2 ³ - 1).
This 76-digit raised number was discovered by Kevin Brown.

The number of divisors is: (126 + 1) 2 ⁶ = 8128.
The sum of the divisors is: (2 ^{126 + 1} -1) 2 ^{61 + 31 + 19 + 7 + 5 + 3} = (2 ¹²⁷ -1) 2 ¹²⁶ .

{\ displaystyle \ tau (60865556 \ dotso) = (2 ^ {7} -1) 2 ^ {6} = 8128}

and are perfect numbers.

{\ displaystyle \ sigma (60865556 \ dotso) = (2 ^ {127} -1) 2 ^ {126}}

It is not known whether there are any other raised numbers or whether there could be an infinite number of or even odd raised numbers.

literature

Clifford A. Pickover : Dr. Googol's wondrous world of numbers . Deutscher Taschenbuch-Verlag, Munich 2005, ISBN 3-423-34177-7 .
Jean-Marie De Koninck: Those Fascinating Numbers. American Mathematical Society , Providence 2009, ISBN 978-0-8218-4807-4 , p. 4 ( online ) and p. 395 ( online ).

Web links

Sublime Numbers on mathpages.com

Individual evidence

↑ Follow A081357 in OEIS

[1] Follow A081357 in OEIS