cp's OEIS Frontend

A331942 a(n) = number of primes of the form P(k) = k^2 + 1 <= 10^n as predicted by the Hardy and Littlewood Conjecture F, rounded to nearest integer. The actual number of primes is A083844(n).

%I A331942 #6 Feb 02 2020 22:23:01
%S A331942 1,4,9,20,48,121,317,855,2356,6609,18787,53970,156385,456404,1340088,
%T A331942 3955219,11726332,34903256,104251560,312353236,938461459,2826668497,
%U A331942 8533343468,25814350227,78239112814,237541788793,722354115787,2199893807666,6708847354653,20485514756657
%N A331942 a(n) = number of primes of the form P(k) = k^2 + 1 <= 10^n as predicted by the Hardy and Littlewood Conjecture F, rounded to nearest integer. The actual number of primes is A083844(n).
%C A331942 Comparison of actual and approximated number of primes < 10^n:
%C A331942    Limit
%C A331942    10^n
%C A331942     | A083844(n)
%C A331942     |   | a(n)
%C A331942     |   | |   (a(n) - A083844(n))/A083844(n)
%C A331942    10^1 2 1  -0.50000
%C A331942    10^2 4 4   0.0
%C A331942    10^3 10 9  -0.10000
%C A331942    10^4 19 20  0.052632
%C A331942    10^5 51 48  -0.058824
%C A331942    10^6 112 121  0.080357
%C A331942    10^7 316 317  0.0031646
%C A331942    10^8 841 855  0.016647
%C A331942    10^9 2378 2356  -0.0092515
%C A331942   10^10 6656 6609  -0.0070613
%C A331942   10^11 18822 18787  -0.0018595
%C A331942   10^12 54110 53970  -0.0025873
%C A331942   10^13 156081 156385  0.0019477
%C A331942   10^14 456362 456404  9.2032E-5
%C A331942   10^15 1339875 1340088  0.00015897
%C A331942   10^16 3954181 3955219  0.00026251
%C A331942   10^17 11726896 11726332  -4.8095E-5
%C A331942   10^18 34900213 34903256   8.7191E-5
%C A331942   10^19 104248948 104251560   2.5055E-5
%C A331942   10^20 312357934 312353236  -1.5040E-5
%C A331942   10^21 938457801 938461459   3.8979E-6
%C A331942   10^22 2826683630 2826668497  -5.3536E-6
%C A331942   10^23 8533327397 8533343468   1.8833E-6
%C A331942   10^24 25814570672 25814350227  -8.5396E-6
%C A331942   10^25 78239402726 78239112814  -3.7054E-6
%C A331942   10^26 237542444180 237541788793  -2.7590E-6
%C A331942   10^27 722354138859 722354115787  -3.1940E-8
%C A331942   10^28 2199894223892 2199893807666  -1.8920E-7
%H A331942 Keith Conrad, <a href="https://kconrad.math.uconn.edu/articles/hlconst.pdf">Hardy-Littlewood Constants</a>, (2003).
%F A331942 b(m) = round (C * Integral_{x=2..m} 1/log(x) dx), with C ~= 0.6864067314..., the Hardy-Littlewood constant for k^2 + 1 (A331941); a(n) = b(10^(n/2)).
%o A331942 (PARI)
%o A331942 C=0.68640673140912300455609634836350943408916655062787977896811707366392;
%o A331942 x=1.0;S10=sqrt(10);for(k=1,30,x*=s10;print1(round(C*intnum(y=2,x,1/log(y))),", "))
%Y A331942 Cf. A083844, A331941.
%K A331942 nonn
%O A331942 1,2
%A A331942 _Hugo Pfoertner_, Feb 02 2020