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4.Behaviour of pmax(b)

base example pmax(b)
2 10 1
3 2^3 3
4 3^3 3
5 3^2*2^40 6
6 5^1*2^7 5
7 3^44*2^1 8
8 7^2*5^4*3^3 6
9 7^1*5^1*2^24 7
10 7^5*3^20*2^4 11
11 5^23*3^11*2^23 12
12 11^9*5^8*2^5 7
13 11^9*7^17*5^7*3^16 14
14 11^1*3^18*2^32 12
15 13^1*11^5*7^14 14
16 7^3*5^2*3^7 8
17 13^14*5^90*2^44 17
18 17^1*5^6*2^6 10
19 7^6*5^59*3^11*2^69 18
20 13^2*11^7*3^1 12
21 19^6*5^1*3^22 14
22 13^3*5^13*3^21 14
23 7^15*5^18*3^46*2^7 19
24 19^3*5^1*2^14 10
25 3^15*2^24 14
26 17^4*3^77*2^26 18
27 19^2*11^4*7^8 12
28 5^13*3^8*2^31 16
29 23^12*11^40*2^77 24
30 7^2*3^12*2^13 13
31 11^118*5^2*2^18 22
32 7^4*3^6 11
33 7^9*3^49*2^65 17
34 31^2*19^35*3^34 20
35 19^15*5^6*2^4 16
36 23^5*19^3*11^3 12
37 13^97*11^6*2^40 24
38 37^22*13^45*2^46 22
39 5^4*3^28*2^69 19
42 37^5*19^8*2^14 17
43 41^46*11^90*3^5 25
64 17^3*5^12 13
81 11^9*5^3*2^6 15
125 7^3*3^10*2^62 19
128 11^3*5^6*2^1 14
243 13^3*3^1*2^50 19
256 11^2*7^6*5^2 15
512 11^6*7^8*3^9 16
625 11^10*3^20*2^28 24
729 13^18*7^4*2^7 21
1024 5^5*3^16 18
2048 7^1*5^5*3^21 19
2187 7^2*5^1*2^42 22
3125 13^11*3^9*2^16 30
4096 11^6*7^10*3^1 22
6561 11^7*3^2*2^7 26
8192 7^6*5^1*3^19 25
15625 11^6*3^16*2^36 36
16384 13^6*11^3*5^10 25
19683 11^2*7^17*5^13 33
32768 17^9*7^2*3^7 28
59049 7^57*5^4*2^25 36
65536 11^2*5^19*3^21 26
131072 19^20*3^16*2^3 24
177147 11^36*7^79 33
262144 17^8*13^29*3^98 25
524288 13^55*7^2*3^1 24
531441 13^54*11^82*2^55 30
1048576 13^15*7^9 27


The table shows a connection between pmax(b) and the numper of divisors of b. Primes reach higher values.
This will be shown in the following table.

primes 2 primefactors 3 primefactors 4 primefactors 2n 3n 5n
2 / 11 4 / 3 8 / 6 16 / 8 1 3 6
3 / 3 6 / 5 12 / 7 24 / 10 3 7 14
5 / 6 9 / 7 18 / 10 36 / 12 6 12 19
7 / 8 10 / 11 20 / 12 8 15 24
11 / 12 14 / 12 27 / 12 11 19 30
13 / 14 15 / 14 28 / 16 13 21 36


You can get more data by downloading an Excel-file.

5.Literature
Last Update: by Sascha Kurz
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