The omega number (n − 1) and its factors

Regarding the number base n, the omega number is simply (n − 1). The factors tα > 2 of (n − 1) are called alpha factors in this work. These factors are integers necessarily coprime to n, but have the brief minimum period 1. In base n, ω = (n − 1) is always written as the largest digit or (natural) numeral of the base, assuming standard positional notation and Hindu-Arabic-like numerals.

The omega number tω is one of the handiest concepts regarding number bases, since omega and its factors have a relatively simple and intuitive divisibility test and a mixed recurrent expansion of 1/tω involving a single recurrent digit. These traits are inherited by products of tω and n-regular numbers r. Decimally, the numbers 3 and 9 are omega; the divisibility rule that applies to them both (the digit sum divisibility test) goes by the name of “the Rule of 3” and “the Rule of 9”.

The map below plots omega factors in blue (tω ■). Number base n appears on the vertical axis and digit k appears on the horizontal. The number 2 in odd bases (tαω ■) is also technically a factor of (n + 1) but is also same of omega = (n − 1). Since the omega divisibility rule is simpler, we can consider 2 in an odd base an omega factor.

The omega number in the multiplication table of base n

In any base n, α is written as the largest numeral ω in that base, and its small multiples kω with 1 ≤ k ≤ (n − 1) are a pair of digits that, if added together, would sum to ω. The pattern is somewhat distinctive and easy to memorize. The square of omega for bases n > 2 is always written “D1”, with D = (n − 2). For example, the decimal 9 × facts are:

9  18  27  36  45  54  63  72  81  90   …

In hexadecimal (base 16), “f” (digit 15) = 3 × 5, thus we see every fifth multiple of 3 has the property that the sum of digits is fifteen:

 3  6  9   c   f  12  15  18  1b  1e  21  24  27  2a  2d  30 …

We likewise observe every third multiple of 5 has the same property:

 5  a  f  14  19  1e  23  28  2d  32  37  3c  41  46  4b  50…

The omega number and its factors as denominators, expanded in base n

Omega (n − 1) is coprime to n, since the smallest prime is 2; no prime factor can be found that divides both n and (n − 1). Omega factors tω> 1 in the denominator 1/tω have a single digit purely recurrent expansion in base n, as the recurrent period of 1/tω is 1. In decimal, 1/9 = .111… and 1/3 = .333…. In nonary (base 9), 1/8 = .111…, 1/4 = .222…, and 1/2 = .444…. In hexadecimal, 1/f = .111…, 1/5 = .333…, and 1/3 = .555…. In any odd base n, one half is expanded to the endlessly repeated digit n/2 rounded down (i.e., floor(n/2)). Therefore in base n = 175, 1/2 would be the radix point followed by endlessly repeating digit 87.

The table below diagrams the behavior of numbers k at the top as unit fractions 1/k, with bold numerals nonrepeating, italic repeating. The omega numbers and their factors tω > 2 appear in light blue (■), while 2 in an odd base appears light purple (■).

  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 1 01 01 0011 001 001 001 000111 00011 0001011101 0001 000100111011 0001 0001 0001
3 1 1 02 0121 01 010212 01 01 0022 00211 002 002 001221 00121 0012
4 2 1 1 03 02 021 02 013 012 01131 01 010323 0102 01 01
5 2 13 1 1 04 032412 03 023421 02 02114 02 0143 013431 013 0124
6 3 2 13 1 1 05 043 04 03 0313452421 03 024340531215 023 02 0213
7 3 2 15 1254 1 1 06 053 0462 0431162355 04 035245631421 03 0316 03
8 4 25 2 1463 125 1 1 07 06314 0564272135 052 0473 04 0421 02
9 4 3 4 17 14 125 1 1 08 07324 06 062 057 053 05
10 5 3 25 2 16 142857 125 1 1 09 083 076923 0714285 06 0625
11 5 37 28 2 19 163 14 124986 1 1 0a 093425a17685 087 08 0762
12 6 4 3 2497 2 186a35 16 14 12497 1 1 0b 0a35186 09724 09
13 6 4 3 27a5 2 1b 18 15a 13b9 12495ba837 1 1 0c 0b36 0a74
14 7 49 37 2b 249 2 1a7 17ac63 158 13b65 1249 1 1 0d 0c37
15 7 5 3b 3 27 2 1d 1a 17 156c4 13b 124936dca5b8 1 1 0e
16 8 5 4 3 2a 249 2 1c7 19 1745d 15 13b 1249 1 1
17 8 5b 4 36da 2e 274e9c 2 1f 1bf5 194adf7c63 17 153fbd 13afd6 1249 1
18 9 6 49 3ae7 3 2a5 249 2 1e73a 1b834g69ed 19 16gb 152a 13ae7 1249
19 9 6 4e 3f 3 2dag58 27 2 1h 1dfa6h538c 1b 18ebd2ha475g 16ehc4 15 13ad
20 a 6d 5 4 36d 2h 2a 248hfb 2 1g759 1d6 1af7dgi94c63 18b 16d 15
21 a 7 5 4 3a 3 2d 27 2 1j
 1f 1cj8 1a 18 16bh
22 b 7 5b 48hd 3e 3 2gb 29h 248hd 2 1i7 1f5 1c 1a5j 185b
23 b 7f 5h 4di9 3j 36d 2k 2chka5 26kg 2 1l 1hfl57 1ei 1c63 1a
24 c 8 6 4j 4 3a6kdh 3 2g 29e 248haljf6d 2 1k795cm3geib 1h3a6kd 1e9 1c
25 c 8 6 5 4 3e7 3 2jb 2c 26kb9 2 1n 1jg 1g 1e
26 d 8h 6d 5 48h 3iem7b 36d 2n 2f 29bl7 248h 2 1m7b3ie 1j 1g6d
27 d 9 6k 5alg 4d 3n 3a 3 2io8 2c79m 26k 2 1p 1lg5a 1if5
28 e 9 7 5gmb 4i 4 3e 3 2mb5g 2f7hmpcka5 29 248h6cpnjalf 2 1o7d 1l
29 e 9j 7 5n 4o 4 3i 36cpmg 2q 2id57qafnl 2c 26k 2 1r 1ng9
30 f a 7f 6 5 48h 3mf 3a 3 2logar85dj 2f 296rkn 248h 2 1q7f
                  ...........   ............   ............... ...........  

Divisibility test for omega numbers and factors

The base-n divisibility test applying to a factor of omega = (n − 1), simply involves adding all the digits of the number together, repeating if necessary (i.e., the digital root). It is well-known decimally that we can determine if a number is divisible by 3 if we add the digits of the number together and the result is clearly a multiple of 3. Example: 121 is not divisible by 3 since 1 + 2 + 1 = 4 is clearly not divisible by 3, but 123 is, since 1 + 2 + 3 = 6, and 6 is clearly a multiple of 3.

In this work we generally term this simple rule “the omega divisibility test”; it is among the simplest divisibility tests and is practical so long as the number of digits to evaluate is reasonable. Decimally the omega test pertains to (10 − 1) = 9 and its factor 3 and is known as “the Rule of 9” and “the Rule of 3” respectively. The omega rule appears in other bases where (n − 1) is composite with small prime factors (most cases, especially in hexadecimal) or is an important yet fairly small prime (as in the case of base 6).

Using hexadecimal (base 16) as an example, we can use the omega rule (add the digits) to determine if a number is divisible by 3, 5, or f (digit fifteen). These rules ought to be mighty useful in hexadecimal, since 16 only has a single distinct prime divisor 2. The omega test in this base extends an ability to discern prime composition of numbers to the smallest three primes {2, 3, 5}.

In hexadecimal, a number x is divisible by 3 if we take the sum of digits of x, repeating as necessary until it is clear that the sum itself is clearly divisible by 3. For example, hexadecimal “81bf1” is divisible by 3 since the sum of digits is “24”; if this isn’t clear (it ought to be from the hexadecimal multiplication table) then we can sum again: 6 is clearly a multiple of 3, so “81bf1” must also be a multiple of 3. (Hexadecimal “81bf1” is the 12th power of 3, 531441, which we can take the digit sum in base 3 to get 18, clearly divisible by 3, thus decimally proven to be a multiple of 3 as well.) In bases 10 and 16, we have the familiar Rule of 3, however in hexadecimal this does not pertain to 9 in base 16, since 9 is not a divisor of (16 − 1)!

In hexadecimal, a number x is divisible by 5 if we take the sum of digits of x, repeating as necessary until it is clear that the sum itself is clearly divisible by 5. As an example we examine the 5th power of 5 expressed hexadecimally as “c35”. Taking the sum of digits, we arrive at “14”, which should be familiar to those who know the hexadecimal multiplication table as the equivalent to decimal 20, clearly a multiple of 5. In contrast, decimally, 5 is regular (more precisely, a divisor) and thus we examine the end digit to see if it is either 0 or 5; this approach does not work with 5 in base 16, since 5 is coprime to 16.

Finally, in hexadecimal, a number x is divisible by “f” (digit fifteen) if we take the sum of digits of x, repeating as necessary until it is clear that the sum itself is clearly divisible by “f”.

The omega test is fairly common across many number bases n, especially those for which (n − 1) is relatively highly factorable or at least composite, with small primes as factors. Since we can consider 2 in an odd base as omega, the omega case is more plentiful for small numbers than the alpha, and given its simplicity, on average the omega would seem more useful in more bases than the alpha. For example, it is especially useful in bases 10 and 16. Users of bases 12 and 14 would see the omega test as a curious but rare and impractical trick that pertains to 11 and 13, respectively.

The decimal omega tests provide a convenient means of testing for 3 and its square, 9, a sort of “quasi-divisibility” considering 3 is the prime “skipped” in the factorization of 10 = 2 × 5. It is a sort of “saving grace” for a semiprime base that might otherwise behave like base 14.

The hexadecimal omega tests are perhaps a runner-up to the decimal, extending the prime-divisibility testing range for base 16 to the smallest three primes. The centovigesimal (base 120) omega tests apply to 7, 17, and 119, while the alpha provides testing for 11 and 121 (the square of 11), extending this flexible base's testing range to the smallest 5 primes and the seventh.

The number 2 in an odd base

The number 2 divides both (n − 1) and (n + 1) for odd n, therefore is technically both and omega and alpha factor. The period for 1/2 expanded in an odd base is 1. The omega rules are simpler though taking the alternating sum in an odd base and determine that a number x is even also works. Thus it is convenient for all practical purposes to consider 2 in an odd base as an omega factor.

The evenness test in an odd base

To determine if an arbitrary integer x is for odd base n, simply take the digit sum. If the sum is clearly even, so is x. Example: the nonary (base 9) number “108807” is even since the sum of its digits is “26”; we can sum again and get “8” which is clearly even.

Omega inheritors

An omega inheritor is a composite number k with at least two distinct prime factors p, with one of the prime power factors pm | k also divides (n − 1).

There are two kinds of omega inheritors, necessarily composite numbers that have an omega factor tω | (n − 1) as a divisor. The omega-regular inheritor involves at least one factor r regular to n and sometimes a factor coprime to n in addition to tω, and is semicoprime to n. The omega-coprime inheritor involves only factors t coprime to n in addition to tω.

We are mostly concerned with simple omega-regular inheritors as the numbers to which they pertain are fairly small and handy.

The simple omega-regular inheritor (sω ■) is a necessarily composite sω = r × tω semicoprime to n such that the coprime factor tω divides (n − 1) and the regular factor r is practical. Armed with a regular divisibility test for r and an omega divisibility test for tω, we can use a compound divisibility test for the alpha inheritor(sω. Decimally, given the Rule of 3 and the decimal evenness test, we can determine whether an arbitrary decimal number x is divisible by 6; this method is familiar to schoolchildren. Example: we know 216 is divisible by 6 since it is even and since (2 + 1 + 6) = 9, which is clearly divisible by 3. In hexadecimal (base 16) there are plenty of simple omega-regular inheritors; we have 6, 12, and 24 inheriting from tω= 3, and 10, 20, and 40 inheriting from 5.

Simple omega coprime inheritors are a product of an omega factor tω and a number t coprime to n for which we have a workable divisibility test. In this work, we highlight the alpha-2 divisibility test and the omega-2 divisibility test as workable divisibility tests. The alpha-2 pertains to factors of (n² + 1), while the omega-2 test pertains to factors of (n² − 1) that are not also factors of neither (n + 1) nor (n − 1). The number k = 15 = 3 × 5 is a compound omega-coprime inheritor in bases 7, 13, 22, 28, etc.; any base n mod k ≡ 7 or 13. In these bases, we combine the omega test for 3 and the alpha-2 test for 5 to arrive at a compound test for 15. Taking the evenness test in an odd base as omega, the number k = 10 = 2 × 5 is a compound omega-coprime inheritor in bases 3, 7, 13, 17, etc.; any base n mod k ≡ 3 or 7. In these bases we combine the evenness test for 2 in an odd base (which is essentially the omega test) and the alpha-2 test for 5. The number k = 26 = 2 × 13 is a compound omega-coprime inheritor in bases 5, 21, 27, 47, etc.; any base n mod k ≡ 5 or 21. We might anticipate that divisibility tests for large numbers may take a back seat to simply using division at some point.

The compound omega-regular inheritor is any sω = r × tω × t semicoprime to n, where t is coprime to n and is a number for which we have a workable divisibility test. The number k = 60 = 2² × 3 × 5 is a compound omega-regular inheritor in base 22, 58, 82, 118, etc.; any base n mod k ≡ 22 or 58. In these bases, we combine the omega test for 3, the regular test for 4, and the alpha-2 test for 5.

The map below plots alpha factors in green (tα ■) and simple omega-regular inheritors in yellow-green (sα ■). Number base n appears on the vertical axis and digit k appears on the horizontal.

The omega inheritors as denominators expanded in base n

The simple omega regular inheritors are sω = r × tω where r is the regular factor and the tω omega factor. These in denominators of fractions 1/sω have mixed recurrent expansions in base k that feature at least one nonrepeating digit and one recurrent digit. The number of nonrepeating digits varies as to the richness ε of the regular factor r of sα, i.e., there are ε nonrepeating digits. For example, in base ten, one twelfth = .08333…, there are 2 nonrepeating digit and a single repeating digit because 12 = r × tω = 4 × 3, 4 | 10² and 3 | (10 − 1).

The table below diagrams the behavior of numbers k at the top as unit fractions 1/k, with bold numerals nonrepeating, italic repeating. The omega numbers and their factors tω > 2 appear in this version of the chart appear blue (■) while tω = 2 in an odd base n appears purple (■). the omega inheritors appear in green (■) but in ochre (■) when tω = 2. Numbers k = 2tα2 appear in beige (■), while k = tα2 × tω appear in light green (■).

Examples: we observe that in base 10, 1/6 = .1666…, in base 16, 1/12 = .1555…, and in base 6, 1/10 = .0333…. Numbers that are omega-coprime inheritors generally have a purely recurrent period of 4; 1/10 in base 7 is .04620462…, 1/15 in base 13 is .0b360b36… (with “b” signifying digit-11).

  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 1 01 01 0011 001 001 001 000111 00011 0001011101 0001 000100111011 0001 0001 0001
3 1 1 02 0121 01 010212 01 01 0022 00211 002 002 001221 00121 0012
4 2 1 1 03 02 021 02 013 012 01131 01 010323 0102 01 01
5 2 13 1 1 04 032412 03 023421 02 02114 02 0143 013431 013 0124
6 3 2 13 1 1 05 043 04 03 0313452421 03 024340531215 023 02 0213
7 3 2 15 1254 1 1 06 053 0462 0431162355 04 035245631421 03 0316 03
8 4 25 2 1463 125 1 1 07 06314 0564272135 052 0473 04 0421 02
9 4 3 4 17 14 125 1 1 08 07324 06 062 057 053 05
10 5 3 25 2 16 142857 125 1 1 09 083 076923 0714285 06 0625
11 5 37 28 2 19 163 14 124986 1 1 0a 093425a17685 087 08 0762
12 6 4 3 2497 2 186a35 16 14 12497 1 1 0b 0a35186 09724 09
13 6 4 3 27a5 2 1b 18 15a 13b9 12495ba837 1 1 0c 0b36 0a74
14 7 49 37 2b 249 2 1a7 17ac63 158 13b65 1249 1 1 0d 0c37
15 7 5 3b 3 27 2 1d 1a 17 156c4 13b 124936dca5b8 1 1 0e
16 8 5 4 3 2a 249 2 1c7 19 1745d 15 13b 1249 1 1
17 8 5b 4 36da 2e 274e9c 2 1f 1bf5 194adf7c63 17 153fbd 13afd6 1249 1
18 9 6 49 3ae7 3 2a5 249 2 1e73a 1b834g69ed 19 16gb 152a 13ae7 1249
19 9 6 4e 3f 3 2dag58 27 2 1h 1dfa6h538c 1b 18ebd2ha475g 16ehc4 15 13ad
20 a 6d 5 4 36d 2h 2a 248hfb 2 1g759 1d6 1af7dgi94c63 18b 16d 15
21 a 7 5 4 3a 3 2d 27 2 1j
 1f 1cj8 1a 18 16bh
22 b 7 5b 48hd 3e 3 2gb 29h 248hd 2 1i7 1f5 1c 1a5j 185b
23 b 7f 5h 4di9 3j 36d 2k 2chka5 26kg 2 1l 1hfl57 1ei 1c63 1a
24 c 8 6 4j 4 3a6kdh 3 2g 29e 248haljf6d 2 1k795cm3geib 1h3a6kd 1e9 1c
25 c 8 6 5 4 3e7 3 2jb 2c 26kb9 2 1n 1jg 1g 1e
26 d 8h 6d 5 48h 3iem7b 36d 2n 2f 29bl7 248h 2 1m7b3ie 1j 1g6d
27 d 9 6k 5alg 4d 3n 3a 3 2io8 2c79m 26k 2 1p 1lg5a 1if5
28 e 9 7 5gmb 4i 4 3e 3 2mb5g 2f7hmpcka5 29 248h6cpnjalf 2 1o7d 1l
29 e 9j 7 5n 4o 4 3i 36cpmg 2q 2id57qafnl 2c 26k 2 1r 1ng9
30 f a 7f 6 5 48h 3mf 3a 3 2logar85dj 2f 296rkn 248h 2 1q7f
                  ...........   ............   ............... ...........  

Color Canon

The colors used to represent omega numbers and factors in this work appear in the table below:

References

Michael De Vlieger, Exploring Number Bases as Tools, ACM Inroads, 2012, Vol. 3, No. 1, 4-12.

LeVeque 1962, Chapter 1, “Foundation”, Section 1−3, “Proofs by Induction”, pages 11−12, specifically: “For every positive integer n, n and n + 1 have no common factor greater than 1” and the ensuing proof.

Marc Renault, Stupid Divisibility Tricks, (See PDF), Section 3.3, Math Horizons (2006). Retrieved in June 2019.

Maths Is Fun, Divisibility Rules, see “9” and “Factors Can Be Useful”. Retrieved in June 2019.

Hardy & Wright 2008, Chapter IX, “The Representation of Numbers by Decimals”, Section 9.6, “Tests for divisibility”, page 147, specifically: “Next 10ν ≡ 1 (mod 9) for every ν, and therefore A1.10s + A2.10(s − 1) + … + As.10 + A(s + 1) ≡ A1 + A2 + … + A(s + 1) (mod 9). A fortiori this is true mod 3. Hence we obtain the well-known rule ‘a number is divisible by 9 (or by 3) if and only if the sum of its digits is divisible by 9 (or by 3)’.”