Euler’s prime number formula with sieve factor function

THE PRIME NUMBER SERIES 27

Euler’s prime number formula is

$n^2+n+41$

whereby $n \in \mathbb{N}$ with $n\ge 0$ .

This formula returns only prime numbers from n=0 to n=39.

From n=40 onwards, the formula provides seemingly arbitrarily alternating composite numbers and prime numbers.

I have added a sieve factor function to Euler’s formula. The result looks like this:

$41+(n^2+n)\prod\limits_{k=0} ^{13} \lceil\sin^2(\frac{n^2+n+41}{k^2+k+41}\pi)\rceil$

This formula returns only prime numbers from n=0 to n=243, 170 of which are different. The rest are repetitions of the prime number 41.

Represented as a set, these are the prime numbers:

{41,251,281,313,347,383,421,461,503,547,593,641,

691,743,797,853,911,971,1033,1097,1163,1231,1301,

1373,1447,1523,1601,1847,1933,2111,2203,2297,

2393,2591,2693,2797,2903,3011,3121,3347,3463,

3581,3701,3823,3947,4073,4201,4463,4597,4733,

4871,5011,5153,5297,5443,5591,5741,6047,6203,

6361,6521,7013,7351,7523,7873,8231,8597,8783,

8971,9161,9547,9743,9941,10141,10343,10753,

11171,11383,11597,11813,12251,12473,12697,

12923,13151,13381,13613,14083,14321,14561,

15541,15791,16553,16811,17333,17597,17863,

18131,18401,18947,19501,20063,20347,20921,

21211,21503,22093,22391,22691,22993,23297,

23603,23911,24533,24847,25163,25801,27431,

27763,28097,28433,28771,29453,30491,30841,

31193,31547,32261,32621,32983,33347,33713,

35573,35951,36713,37097,37483,37871,38261,

38653,39047,39443,39841,40241,41047,41453,

42683,44351,44773,45197,46051,48221,48661,

49103,49547,49993,50441,50891,51343,51797,

52253,52711,53171,53633,54563,55501,56923,

57881,58363,59333}

Munich, 29 May 2024

Gottfried Färberböck

THE PRIME NUMBER SERIES