May 23, 2014

Horse 1680 - The Search for the First Boring Number

All the way back in Horse 1193 I asserted that 14 was the first boring number. In a series of tweets with Adam Spencer, I thought that 26 was the first boring number. It turns out that 14 is actually quite interesting and that 26 is also interesting.
Of course being the proper studious nerdulent geek that I am, the question bothered me enough to try and work out what the first boring number is.
The Interesting Number Paradox suggests that the quest is pointless because as soon as you found the first uninteresting number, then it is itself is therefore interesting because it is the smallest uninteresting number and thus contradicts itself.
I say this is bunk. The first boring number, must be the first one which isn't part of an interesting series. Like the Sieve of Eratosthenes which is a useful tool to find primes, there must be an equivalent Boring Number Sieve, which looks at all interesting number sieves.
The game is on!

1 - 1 is its own Square, Cube, Fourth... Infinitieth power of itself and it's also it's own Square Root, Cube Root, Zeroeth root of everything. 1 is a Factorial Number (1!=1), is a Bell Number, a Centered Hexagonal Number, a Happy Number and a Fibonacci Number. 1 is the starting point of most series and for that reason is possibly the most interesting number of all.

2 - 2 is the only even Prime. 2 is a Factorial Number (2!=2) and a Fibonacci Number (1+1=2). 2 is also a Bell Number which I'll explain at some point further down the list.

3 - 3 is a Triangular Number (1+2=3). 3 is a Prime Number. 3 is a Fibonacci Number (1+2=3).
A Prime Number is any number which has no positive divisors other than 1 and itself. 3 is Prime because its only factors are 3 and 1.

4 - 4 is a Square Number (2² = 4). 4 is a Semiprime Number (2 & 2).
A Semiprime number has only factors other than one, which are also prime. A semiprime number has no composite factors. 33 for instance has factors other than 33 of 3 and 11 which are both prime. 4 has factors of 2 and 2, which are also both prime.

5 - 5 is Prime. 5 is a Fibonacci Number (2+3=5). 5 is a Square Pyramid Number (1²+2²=5)
Square Pyramid Numbers - Let's make a pyramid out of chocolate squares (because that's the yummiest kind of pyramid). In the top layer we put one square of chocolate (1²), in the next layer four (2²), in the next layer nine (3²)... oh I love yummy maths.

6 - 3 is a Triangular Number (1+2+3=6). 6 is a Semiprime Number (2x3=6). 6 is a Factorial (3!=6). 6 is a Perfect Number.
How can a number be Perfect? A Perfect Number is one whose positive divisors when added together, produce a sum equal to that number. 6 has factors of 1, 2 amd 3 and 1+2+3=6 which is perfect.

7 - 7 is Prime. 7 is a Centred Hexagonal Number. 7 is Happy.
7 isn't Happy because it's lucky for some.
Start with 7; find the sum of the square of its digits which is 7² = 49. Keep going and if that that process ends in 1, then the number is happy. If not... it's a sad sad number.
7
7² = 49
4² + 9² = 97
9² + 7² = 130
1² + 3² + 0² = 10
1² + 0² = 1 yay! 7 is Happy!

8 - 8 is Cubic Number (2³=8). 8 is a Fibonacci Number (3+5=8).
Leonardo Pisano Bigollo who went by the stage name of Fibonacci, wanted to know the logical population growth of rabbits. By looking at pairs of rabbits he suggest that they should increase according to the series:
1 1 2 3 5 8 13 21 34 55... where the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month.
Fibonacci obviously had a problem with rabbits. Maybe he should have asked the Chinese Emperor Nasi Goreng what to do about the problem.

9 - 9 is a Square Number (3²=9). 9 is a Semiprime Number (3 & 3).

10 - 10 is a Triangular Number (1+2+3+4=10). 10 is a Happy Number. (1² + 0² = 1 yay! 10 is Happy!) 10 is also a Harshad number (1+0 = 1 which is a factor of 10)
Actually, all the numbers from 1 to 10 in any base are Harshad Numbers. A Harshad Number is one whose digits add up to a factor of the number in question. It would be easy to be tempted for instance to assume that all multiples of 9 are Harshad numbers because of the divisibility test for 9 (see Horse 1676) but for 99, 9+9=18 and 18 is not a factor of 99.

11 - 11 is Prime. 11 is also the only prime number as far as I know which the digits repeat.

12 - 12 is a Harshad number (1+2 = 3 which is a factor of 12). 12 is also an Abundant Number.
Abundant Numbers are those whose factors add together to give a total bigger than the number in question.  12's factors are 1 + 2 + 3 + 4 + 6 = 16. Numbers which are not Abundant are Deficient and therefore not interesting. It took me forever to work this out but I think that the smallest abundant number which is odd is 945, but I'm not sure. I'm probably wrong.
Twelve is also the largest number with a single morpheme. Thir-teen which is next, is a compound name.

13 - 13 is Prime. 13 is also a Permutable Prime (with 31). 13 is a Fibonacci Number (5+8=13).
13 as a Permutable Prime has a cousin; who is 31. In base-10, Permutable Primes can only exist which contain only the numbers 1 3 7 and 9, since the only other primes which exist and do not contain these integers are 2 & 5. All numbers ending in 2 are divisible by 2 and numbers ending in 5 are divisible by 5; so apart from 2 & 5, they're all complex and therefore not interesting.

14 - 14 is Semiprime (2x7=14). 14 is a Square Pyramid Number (1²+2²+3²=14).

15 - 15 is a Triangular Number (1+2+3+4+5=15). 15 is Semiprime (3x5=15).

16 -16 is a Square Number (4²=16). 16 is also the fourth power of two.

17 - 13 is a Permutable Prime (with 71).

18 - 18 is a Harshad number (1+8 = 9 which is a factor of 18). 12 is an Abundant Number (1+2+3+6+9=21).

19 - 19 is Prime. 19 is a Centered Hexagonal number. 19 is a Happy Number.
19
1² + 9² = 82
8² + 2² = 68
6² + 8² = 100
1² + 0² + 0² = 1 yay!

20 - 20 is an Abundant Number (1+2+4+5+10=22). 20 is a Harshad number (2+0 = 2 which is a factor of 20).

21 - 21 is a Triangular Number (1+2+3+4+5+6=21). 20 is a Harshad number (2+1 = 3 which is a factor of 21). 13 is a Fibonacci Number (8+13=21).

22 - 22 is Semiprime (2x11=22)

23 - 23 is Prime.

24 - 24 is an Abundant Number (1+2+3+4+6+8+12=36).  6 is a Factorial (4!=24). 24 is a Harshad number (2+4= 6 which is a factor of 24).

25 - 25 is a Square Number (5²=25). 25 is Semiprime (5x5=25).

26 - 26 is Semiprime (2x13=26). 26 is also a number between a square and a cube (25 & 27) but nobody knows if that's unique or not. 26 is also a Permutable Semiprime (with 62).

27 - is Cubic (3³=27). 27 is a Harshad number (2+7 = 9 which is a factor of 27).

28 - is a Perfect Number (1+2+4+7+14=28). 28 is a Happy Number.
28
2² + 8² = 68
6² + 8² = 100
1² + 0² + 0² = 1 yay!

29 - 29 is Prime.

30 - 30 is a Square Pyramid Number (1²+2²+3²+4²=30). 30 is a Harshad number (3+0= 3 which is a factor of 30).

31 - 31 a Permutable Prime (with 13).

32 - 32 is the fifth power of two.

33 - 33 is Semiprime (3x11=33).

34 - 34 is Semiprime (2x17=34).  34 is a Fibonacci Number (13+21=34).

35 - 33 is Semiprime (5x7=35).

36 - 36 is an Abundant Number (1+2+3+4+6+9+12=37). 36 is a Harshad number (3+6 = 9 which is a factor of 36).

37 - 37 is a Permutable Prime (with 73).

38 - 38 is Semiprime (2x19=38).

39 - 39 is Semiprime (3x13=39).

40 - 40 is a Harshad number (4+0 = 4 which is a factor of 40). 40 is an Abundant Number (1+2+4+5+8+10+20=50).

41 - 41 is Prime.

42 - 42 is a Harshad number (4+2 = 6 which is a factor of 42). 42 is an Abundant Number (1+2+3+6+7+14+21=54).

43 - 43 is Prime.

44 - 44 is a Happy Number.
44
4² + 4² = 32
3² + 2² = 13
1² + 3² = 10
1² + 0² = 1 yay!

45 - 45 is a Triangular Number (1+2+3+4+5+6+7+8+9 = 45). 45 is a Harshad number (4+5 = 9 which is a factor of 45).

46 - 46 is Semiprime (2x23=46).

47 - 47 is Prime.

48 - 48 is an Abundant Number (1+2+3+4+6+12+24=52). 48 is a Harshad number (4+8 = 12 which is a factor of 48).

49 - 49 is a Square Number (7²=49). 49 is Semiprime (7x7=49).

50 - 50 is a Harshad number (5+0 = 5 which is a factor of 50).

51 - 51 is Semiprime (3x17=51).

52 - 52 is a Bell Number.
Bell Numbers count the number of different ways to partition a set. 52 is the 5th Bell Number and so counts the number of ways to count off the number of possible sets for 5 things.
- All the singles.
- All the possible pairs.
- All the possible triples.
- All the possible quad groups.
- All the possible quint groups (which for 5 things is 1).
- All the possible two pairs.
- All the possible full houses (a pair and a triple).

Rather than bother to count them up, simple start a Bell Triangle.
Copy the last term to start a new row and for every new term add downwards; then place the result next to it. Bell Numbers appear at both ends of a row.
 1
 1   2
 2   3   5
 5   7  10  15
15  20  27  37  52
52  67  87  114 151 203

Simples.

53 - 53 is Prime.

54 - 54 is a Harshad number (5+4 = 9 which is a factor of 54). 54 is an Abundant Number (1+2+3+6+9+18+27=66).

55 - 55 is a Triangular Number (1+2+3+4+5+6+7+8+9+10=55). 55 is a Square Pyramid Number (1²+2²+3²+4²+5²=55). 55 is Semiprime (5x11=55). 55 is a Fibonacci Number (21+34=55).

56 - 56 is an Abundant Number (1+2+4+7+8+14+28=64).

57 - 57 is Semiprime (3x19=57).

58 - 58 is Semiprime (2x29=58).

59 - 59 is Prime.

60 - 60 is a Harshad number (6+0 = 6 which is a factor of 60). 60 is an Abundant Number (1+2+3+4+5+12+15+20+30=92).

61 - 61 is Prime. 61 is a Centred Hexagonal Number.

62 - 62 is Semiprime (2x31=62). 62 is also a Permutable Semiprime (with 26).

63 - 63 is a Harshad number (6+3 = 9 which is a factor of 63).

64 - 64 is a Cubic number (4³=64). 64 is a Square number (8²=64). 64 is the sixth power of two.

65 - 65 is Semiprime (3x13=65).

66 - 66 is a Triangular Number (1+2+3+4+5+6+7+8+9+10+11=66).

67 - 67 is Prime.

68 - 68 is a Happy Number.
68
6² + 8² = 100
1² + 0² + 0² = 1 yay!

69 - 69 is Semiprime (3x23=69)

70 - 70 is a Harshad number (7+0 = 7 which is a factor of 70). 70 is a Happy Number.
70
7² + 0² = 49
4² + 9² = 97
9² + 7² = 130
1² + 3² + 0² = 10
1² + 0² = 1 yay!

71 - 71 is a Permutable Prime (with 17).

72 - 72 is a Harshad number (7+2 = 9 which is a factor of 72). 72 is an Abundant Number (1+2+3+4+6+8+9+12+18+24+36=123).

73 - 73 is a Permutable Prime (with 37).

74 - 74 is Semiprime (2x37=74)

75 - 75... er... 75... 75? Hello?!

At this point my brain collapsed because I couldn't find anything that 75 was. 75 was sufficiently uninteresting enough to be called the first boring number in my sieve.
Except now I find out that 75 is a pentagonal pyramidal number, a Keith number and a
Colombian number.

ARGH!

Is there no end to this madness?!

Maybe.

A chap called Neil Sloane invented a thing called the On-Line Encyclopedia of Integer Sequences or OLEIS (Link: http://oeis.org/). It has a whole host of number lists; so maybe a boring number would be the first which doesn't appear in those lists.

Alternatively, an irrational number which is even more irrational than √2 or π might very well be boring simply because it's not particularly useful for anything. We care about π and √2 because they have some useful meaning to us. Even just between the integers of 0 and 1 there are an infinite amount of equally boring un-useful numbers that we simply don't care about.
The first boring number is maybe just some nebulously small number, infinitesimally larger than 0, just hanging about minding its own business; safe in the knowledge that it won't be found because nobody cares about it.
Of course, I've just been told that Maths itself is boring; so ALL numbers are boring; so I suppose it depends who you ask anyway.

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