20220627, 15:09  #1 
Dec 2008
you know...around...
3·269 Posts 
HardyLittlewood constants for quadratic polynomials
The polynomial x^{2}+x+a with integer x and odd integer a has a certain density of prime numbers which can be measured by an associated HardyLittlewood constant, HL(x^{2}+x+a).
Example: HL(x^{2}+x+1) = 2.241465507..., loosely speaking that x^{2}+x+1 has a probability about 2.24 times as high to be prime than a random number of the same size. Did I get that right that for every r that divides 4a1 and every nonnegative integer n, HL(x^{2}+x+ (1+(4a1)*r^{2n})/4) is equal? What broader context can this be viewed in? 
20220712, 21:37  #2 
Dec 2008
you know...around...
807_{10} Posts 
I keep up until I hit a raw nerve
Followup question:
Is there a similar measure for the numbers of the form 2kp+1 yielding a lot of composites, in which case the generalized repunits (b^p1)/(b1) would produce an aboveaverage number of primes? Successive records of this form should be* formed by the primes p = {3, 5, 7, 13, 17, 19, 31, 59, 109, 157, 167, 317, 457, 521, 1163, 1741, 1997, 2053, 3079, 3833, 5227, 5641, 11069, 12919, 13469, 14419, 16103, 19813, 19891, 22441, 22691, 28229, 30391, 31667, 37189, 39097, 39829, 51413, 71593, 74507, 85627, 93607, 104801, 117899, 138163, 170167, 216091, ...} * not all numbers in the list may stand the test of a more sophisticated computation, but I'm confident that most of them will 
20220713, 01:37  #3  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7^{2}×73 Posts 
Quote:
However, (b^1271)/(b1) also produce a muchaboveaverage number of primes, but 4*127+1 is prime, I really do not know why. Indeed, (b^p1)/(b1) would produce an aboveaverage number of primes for these primes, and a belowaverage number of primes for primes p such that 2*p+1 is prime (SophieGermain primes) (since (b^p1)/(b1) is divisible by 2*p+1 if b is a quadratic residue mod 2*p+1 and b != 1 mod 2*p+1), primes p such that 4*p+1 is prime, primes p such that 6*p+1 is prime You can see https://oeis.org/A066180, which is the smallest base b such that the generalized repunit (b^p1)/(b1) is prime, it will be belowaverage number for the primes in https://oeis.org/A035095, and aboveaverage number for the Sophie Germain primes, indeed, 2^5211 is prime, and 521*n+1 is composite for all n<32, also, in both base 10 and base 12, the repunit R317 is prime, and 317*n+1 is composite for all n<30, it is also notable that there are many bases such that the repunits R17, R19, R31 is prime, but not for R59, since 59 is an irregular prime, also aboveaverage number for the irregular primes (https://oeis.org/A000928), e.g. for p=37, the smallest irregular prime, https://oeis.org/A066180 is 61, which is much aboveaverage, another case is p=491, which combined with the fact that it is a Sophie Germain prime and an irregular prime (it is also notable that it is the smallest Sophie Germain irregular prime whose corresponding safe prime is a regular prime), https://oeis.org/A066180 is 514, which is much aboveaverage, note that the two primes p which Mersenne wrongly said 2^p1 is prime, 67 and 257, are both irregular primes, and all primes p < 257 such that 2^p1 is prime are regular primes. Last fiddled with by sweety439 on 20220713 at 01:40 

20220713, 21:38  #4 
Dec 2008
you know...around...
3·269 Posts 
Many thanks for your reply!
I had devised a scheme some years ago to compare how prolific the generalized repunits are w.r.t. the values of p, by taking a sum \(s_r:=\sum_{q=7,\,q\,prime}^x \sum_{r:\,every\,odd\,prime\,factor\,of\,q1} \frac{r1}q\) and then looking at \(\lim_{x\to\infty} s_r\log\log x\). s_{3} ~ log log x  0.7137809590188862582393 s_{5} ~ log log x  0.8353377999490359325575 s_{7} ~ log log x  1.1224508552674277645804 s_{11} ~ log log x  0.9713749289063147467859 s_{13} ~ log log x  1.2082293777700878099127 s_{17} ~ log log x  1.3685790169746073220730 s_{19} ~ log log x  1.6194280070050371094275 s_{23} ~ log log x  1.0578200394879716206056 s_{29} ~ log log x  1.1685981976748223373826 s_{31} ~ log log x  1.7000384501264852200143 etc. The lower the asymptotic value of s_{p} (or, the larger the value that's being subtracted from log log x), the more likely it is for (b^{p}1)/(b1) to yield a lot of primes. I'm not entirely sure whether this approach is justified, I'd have to dig a little deeper into the theory and also do some number crunching. s_{127} has only the second lowest asymptotic value of all s_{p} with p <= 127, so it wouldn't surprise me if the prime output is quite high. Note that, for instance, 2*k*127+1 is composite for 3 <= k <= 8. Last fiddled with by mart_r on 20220713 at 21:41 
20220713, 23:55  #5  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7^{2}×73 Posts 
Quote:
I indeed have a data for the 2<=b<=4096 such that Phi(n,b) is prime (ignore the numbers with "*", "**", or "***", and plus the prime Phi(6,2), I make this data is for the generalized unique primes), there will a belowaverage number of primes Phi(n,b) if n+1 is prime, or 2*n+1 is prime, or 3*n+1 is prime, or 4*n+1 is prime, and an aboveaverage number of primes Phi(n,b) if k*n+1 is composite for all small k (see https://oeis.org/A034694 and https://oeis.org/A034693 and https://oeis.org/A120857), also, you can see the sequence https://oeis.org/A085398, a(n) will be large if n+1 is prime, or 2*n+1 is prime, or 3*n+1 is prime, or 4*n+1 is prime, and small (e.g. a(n) = 2 or 3) if k*n+1 is composite for all small k (a(n) is usually (but not always) not perfect power (I call the n such that a(n) is perfect power as "unusual numbers", such numbers n are 20, 28, 44, 66, 74, 87, 92, 96, 104, 138, 140, 148, 152, 156, 166, 178, 182, 189, 204, 210, 232, 249, 250, 264, 268, 292, 298, 300, ...), and if a(n) = m^r with r>1, then all prime factors of r divide n, and a(n*d) = m^(r/d) for all d dividing r), a(n) ~ gamma*eulerphi(n) (I conjectured), and I called the n such that a(n) > eulerphi(n) as "Satan numbers", such numbers n are 1, 2, 25, 37, 44, 68, 75, 82, 99, 115, 119, 125, 128, 159, 162, 179, 183, 188, 203, 213, 216, 229, 233, 243, 277, 289, 292, 301, 309, 316, 323, 329, 362, 376, 391, 397, 421, 423, 427, 432, 435, 438, 442, 443, 444, 459, 466, 473, 484, 491, 494, ... Last fiddled with by sweety439 on 20220714 at 00:04 

20220714, 20:12  #6 
Dec 2008
you know...around...
3·269 Posts 
OK, thanks!
It's interesting to note that the prime terms in A120857 are similar to those I gave in post # 2. Maybe compute more terms of A120857 and see how many more coincide with my list? Until then, I'm going to need some time to digest all about cyclotomic polynomials... P.S.: HL(x^{2}+x+a(n)) is the same for all terms a(n) in A066443. Just in case anyone thinks this is worthy to add in the comments. 
20220715, 06:40  #7 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7^{2}×73 Posts 
Does HardyLittlewood constant also exist for exponential sequences (a*b^n+c)/gcd(a+c,b1) (with fixed integers a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1, and variable integer n>=1)?

20220715, 19:35  #8 
Dec 2008
you know...around...
1100100111_{2} Posts 
Maybe so. But those would depend on two or three different variables, I reckon that they would be much harder to calculate. I don't yet have the prerequisites to figure out efficient ways to do that, certainly others already did a lot of work on that, unfortunately I don't know where to start looking for it (at least for the special form of numbers you gave).

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