Number Theory/Relative To Record

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) Numbers for which d(n) sets a new record are called highly composite number 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