Strange number

In mathematics , a natural number n is called a strange number if it fulfills the following two properties:

• It's an abundant number . The sum of its real divisors (all divisors except for the number n itself) is thus greater than the number n itself (for the divisor sum function, or applies ). ${\displaystyle \sigma (n)>2n}$ ${\displaystyle \sigma ^{*}(n)>n}$
• It is not a pseudo-perfect number , that is, it can not be represented as the sum of several different real divisors.

In other words:

The sum of the real divisors (including 1, but without n ) must be greater than the number n , but no subset of these divisors may add up to the number n .

Examples

Example 1: The number 70 has the real divisors 1, 2, 5, 7, 10, 14 and 35. Your real divisional sum is therefore 1 + 2 + 5 + 7 + 10 + 14 + 35 = 74> 70 and the number thus abundant. But you can never sum up the numbers 1, 2, 5, 7, 10, 14 and 35 in such a way that the number 70 comes out. So, the number 70 is a strange number.

Example 2: The number 72 has the real divisors 1, 2, 3, 4, 6, 8, 9, 12, 18, 24 and 36. So your real divisional sum is 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 = 123> 72 and the number thus also abundant. But you can create a sum from the numbers 1, 2, 3, 4, 6, 8, 9, 12, 18, 24 and 36 in such a way that the number 72 comes out, namely 12 + 24 + 36 = 72 (or also 2 + 4 + 6 + 24 + 36 = 72, and there are other options). So the number 72 is not a strange number.

Example 3: The number 74 has the real divisors 1, 2 and 37. Its real divisor sum is therefore 1 + 2 + 37 = 40 <74 and the number is therefore deficient and not abundant. So 74 doesn't even have the first property, so it's not a strange number.

The first strange numbers are the following:

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, ... sequence A006037 in OEIS

properties

• There are an infinite number of strange numbers.
• Let be a positive integer, a prime number with and also a prime number with . Then: ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle Q\in \mathbb {P} }$ ${\displaystyle Q>2^{k}}$ ${\displaystyle R={\frac {2^{k}Q-(Q+1)}{(Q+1)-2^{k}}}\in \mathbb {P} }$ ${\displaystyle R>2^{k}}$

${\displaystyle n=2^{k-1}QR}$ is a strange number.

(with this formula he found the big strange number ) ${\displaystyle n=2^{56}\cdot (2^{61}-1)\cdot 153722867280912929\approx 2\cdot 10^{52}}$

• Let be a strange number and a prime number with , so let p be greater than the sum of all divisors of (including itself). Then: ${\displaystyle n}$ ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle p>\sigma (n)}$ ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle p\cdot n}$ is a strange number.

Example 1:

Be a strange number (with ) and . Then actually is a strange number. ${\displaystyle n=70}$ ${\displaystyle \sigma (n)=1+2+5+7+10+14+35+70=144}$ ${\displaystyle p=151>\sigma (n)=144}$ ${\displaystyle p\cdot n=70\cdot 151=10570}$

Example 2:

Be a strange number (with ) and . Then actually is a strange number. ${\displaystyle n=836}$ ${\displaystyle \sigma (n)=1+2+4+11+19+22+38+44+76+209+418+836=1680}$ ${\displaystyle p=1693>\sigma (n)=1680}$ ${\displaystyle p\cdot n=836\cdot 1693=1415348}$

This results in the following definition: A number is called a primitive strange number if it is not a multiple of another strange number. ${\displaystyle n}$

The first primitive strange numbers are the following:

70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 243892, 254012, 338572, 343876, 388076, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1713592, 1901728, 2081824, 2189024, 3963968, 4128448, ... sequence A002975 in OEIS

There are only 24 primitive strange numbers that are less than a million (but there are 1765 strange numbers that are less than a million).

Unsolved problems

• Are there odd odd numbers? If so, then it has to be. ${\displaystyle >10^{21}}$
• Are there infinitely many strange primitive numbers? It was designed by Giuseppe Melfi already shown that when Cramér's conjecture ( s true) that it follows that there are infinitely many primitive odd numbers.

literature

• József Sándor , Dragoslav Mitrinović , Borislav Crstici: Handbook of Number Theory I . 2nd Edition. Springer-Verlag, Dordrecht 2006, ISBN 1-4020-4215-9 , pp. 113-114 .
• SJ Benkoski, Paul Erdős : On Weird and Pseudoperfect Numbers . In: Mathematics of Computation . tape 28 , 1974, p. 617-623 .

Web links

• József Sándor , Dragoslav Mitrinović , Borislav Crstici: Handbook of Number Theory I. (PDF) Springer-Verlag, pp. 113–114 , accessed on May 21, 2018 (English). 
• SJ Benkoski, Paul Erdős : On Weird and Pseudoperfect Numbers. (PDF) Mathematics of Computation , pp. 617–623 , accessed on May 24, 2018 (English). 
• Eric W. Weisstein : Weird Number . In: MathWorld (English).
• Weird Number . In: PlanetMath . (English)

Individual evidence

1. ↑ József Sándor , Dragoslav Mitrinović , Borislav Crstici: Handbook of Number Theory I . 2nd Edition. Springer-Verlag, Dordrecht 2006, ISBN 1-4020-4215-9 , pp. 113-114 .
2. ↑ Sidney Kravitz: A search for large weird numbers . In: Journal of Recreational Mathematics . tape 9 , 1976, p. 82-85 .
3. ↑ SJ Benkoski, Paul Erdős : On Weird and pseudo Perfect Numbers . In: Mathematics of Computation . tape 28 , 1974, p. 617-623 .
4. ^ Neil Sloane : Weird numbers: abundant but not pseudoperfect - Comments. Retrieved May 24, 2018 . 
5. ↑ Odd Weird Search. rechenkraft.net, accessed on May 25, 2018 . 
6. ^ Giuseppe Melfi : On the conditional infiniteness of primitive weird numbers . In: Journal of Number Theory . tape 147 , 2015, p. 508-514 .