I've made mention of the Euler polynomial
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Figure 1 |
A228183 | Semiprimes generated by the Euler polynomial |
1681, 1763, 2021, 2491, 3233, 4331, 5893, 6683, 6847, 7181, 7697, 8051, 8413, 9353, 10547, 10961, 12031, 13847, 14803, 15047, 15293, 16043, 16297, 17071, 18673, 19223, 19781, 20633, 21797, 24221, 25481, 26123, 26447, ...
This reminded me that the Euler polynomial is not only an excellent producer of primes, at least initially, but also an excellent generator of semiprimes. Up until
If we extend our range to one million, these percentages alter significantly. We find that there are 261080 primes and 458947 semiprimes and thus primes constitute 26.1% while semiprimes constitute 45.9%. About 28% of the numbers generated clearly have three or more factors and, as mentioned earlier, the run of primes and semiprimes stops with \(n=420) because here we encounter the first sphenic number, that is a number with three distinct prime factors. See Figure 2.
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Figure 2 |
Looking at Figure 2, it can be seen that while the primes and semiprimes still abound, the first two sphenic numbers make their appearance when
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Figure 3 |
The first number with four distinct prime factors occurs with
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Figure 4 |
The first number with five distinct prime factors does not occur until
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Figure 5 |
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Figure 6 |
- 31984 prime numbers
- 47937 semiprimes
- 17741 sphenic numbers
- 1661 numbers with four distinct prime factors
- 18 numbers with five distinct prime factors
- 659 numbers that are not square free.
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Figure 7 |
As can be seen in Figure 7, the Euler polynomial does a great job in churning out primes and semiprimes.