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Number Theory/Relative To Record

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In a fluctuating integer sequence, every term has a relative-to-record score, defined by the following formula:

"Relative to Record" =

where S(x) is the observed term of the sequence, and S(m) is the maximum value of S(k) for the range 1 ≤ k ≤ x − 1.

The inverse value is defined by subtracting this value from one.

Both "Relative to Record" and "Relative to Record (Inverse)" are real numbers between 0 and 1. "Relative to Record" is greater than 0.5 if and only if S(x) is a record term in the sequence. "Relative to Record (Inverse)" is less than 0.5 if and only if S(x) is a record term in the sequence. A term S(x) is a record term iff for all positive integers k such that k < x, S(k) < S(x). By convention:

  • The first term of any integer sequence is considered to be a record term.
  • Terms that tie, but don't break, a record are not considered to be record terms.

An integer sequence S(x) is a fluctuating integer sequence iff it is neither "eventually strictly increasing" nor "eventually strictly decreasing." Equivalently:

  • I) There are infinitely many natural numbers n such that S(n + 1) > S(n).
  • II) There are infinitely many natural numbers n such that S(n + 1) < S(n).

In number theory, two important fluctuating integer sequences are the divisor function d(n) and the prime gaps. For any natural number n, d(n) is the number of positive divisors of n. For example, d(15) = 4 because fifteen has four divisors: 1, 3, 5, and 15. The prime gaps, as the name suggests, are the arithmetic differences between consecutive prime numbers. There are 25 prime numbers between 1 and 100, those being 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. To find a prime gap, you subtract a prime from its successor; for example, to find the eighth prime gap, subtract the eighth and ninth primes: 23 − 19 = 4. The first 24 prime gaps are:

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8.

The records in this sequence are g(1) = 1, g(2) = 2, g(4) = 4, g(9) = 6, and g(24) = 8. The next time the record breaks is the thirtieth term: g(30) = 127 − 113 = 14. After that, you won't find a larger term until the ninety-ninth term: g(99) = 541 − 523 = 18. The record breaks yet again at g(154) = 907 − 887 = 20. Due to the prime number theorem, prime gaps grow arbitrarily large, so this record will keep being broken.

Relative to Record (Divisor function)

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Numbers for which d(n) sets a new record are called highly composite numbers. The highly composite numbers below 1000 are:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, and 840.

For all n > 1, d(n) is always at least two. Those numbers with exactly two divisors are the prime numbers.

n d(n) Relative To Record Relative To Record (Inverse)
1 1 N/A N/A
2 2 0.666667 0.333333
3 2 0.500000 0.500000
4 3 0.600000 0.400000
5 2 0.400000 0.600000
6 4 0.571429 0.428571
7 2 0.333333 0.666667
8 4 0.500000 0.500000
9 3 0.428571 0.571429
10 4 0.500000 0.500000
11 2 0.333333 0.666667
12 6 0.600000 0.400000
13 2 0.250000 0.750000
14 4 0.400000 0.600000
15 4 0.400000 0.600000
16 5 0.454545 0.545455
17 2 0.250000 0.750000
18 6 0.500000 0.500000
19 2 0.250000 0.750000
20 6 0.500000 0.500000
21 4 0.400000 0.600000
22 4 0.400000 0.600000
23 2 0.250000 0.750000
24 8 0.571429 0.428571
25 3 0.272727 0.727273
26 4 0.333333 0.666667
27 4 0.333333 0.666667
28 6 0.428571 0.571429
29 2 0.200000 0.800000
30 8 0.500000 0.500000
31 2 0.200000 0.800000
32 6 0.428571 0.571429
33 4 0.333333 0.666667
34 4 0.333333 0.666667
35 4 0.333333 0.666667
36 9 0.529412 0.470588
37 2 0.181818 0.818182
38 4 0.307692 0.692308
39 4 0.307692 0.692308
40 8 0.470588 0.529412
41 2 0.181818 0.818182
42 8 0.470588 0.529412
60 12 0.545455 0.454545
80 10 0.454545 0.545455
100 9 0.428571 0.571429
120 16 0.571429 0.428571
150 12 0.428571 0.571429
300 18 0.473684 0.526316
301 4 0.166667 0.833333
302 4 0.166667 0.833333
303 4 0.166667 0.833333
304 10 0.333333 0.666667
305 4 0.166667 0.833333
306 12 0.375000 0.625000
307 2 0.090909 0.909091
308 12 0.375000 0.625000
360 24 0.545455 0.454545
720 30 0.555556 0.444444
840 32 0.516129 0.483871
850 12 0.272727 0.727273
1,000 16 0.333333 0.666667
2,000 20 0.333333 0.666667
2,500 15 0.272727 0.727273
2,520 48 0.545455 0.454545
2,521 2 0.040000 0.960000
2,522 8 0.142857 0.857143
2,523 6 0.111111 0.888889
3,000 32 0.400000 0.600000
4,000 24 0.333333 0.666667
5,000 20 0.294118 0.705882
5,040 60 0.555556 0.444444
5,041 3 0.047619 0.952381
5,042 4 0.062500 0.937500
5,043 6 0.090909 0.909091
5,044 12 0.166667 0.833333
5,045 4 0.062500 0.937500
5,046 12 0.166667 0.833333
6,000 40 0.400000 0.600000
6,007 2 0.032258 0.967742
6,008 8 0.117647 0.882353
7,000 32 0.347826 0.652174
7,500 30 0.333333 0.666667
7,560 64 0.516129 0.483871
7,561 2 0.033333 0.966667
7,562 8 0.111111 0.888889
7,563 4 0.058824 0.941176
7,564 12 0.157895 0.842105
7,565 8 0.111111 0.888889
7,566 16 0.200000 0.800000
7,567 8 0.111111 0.888889
7,568 20 0.238095 0.761905
7,569 9 0.123288 0.876712
7,570 8 0.111111 0.888889
7,571 4 0.058824 0.941176
7,572 12 0.157895 0.842105
7,573 2 0.033333 0.966667
8,000 28 0.304348 0.695652
9,000 48 0.428571 0.571429
10,000 25 0.280899 0.719101
15,120 80 0.526316 0.473684
16,000 32 0.285714 0.714286
20,000 30 0.272727 0.727273
30,000 50 0.342466 0.657534
50,000 30 0.230769 0.769231
50,400 108 0.519231 0.480769
52,000 48 0.307692 0.692308
60,000 60 0.333333 0.666667
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