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).

1, 4, 9, 20, 48, 121, 317, 855, 2356, 6609, 18787, 53970, 156385, 456404, 1340088, 3955219, 11726332, 34903256, 104251560, 312353236, 938461459, 2826668497, 8533343468, 25814350227, 78239112814, 237541788793, 722354115787, 2199893807666, 6708847354653, 20485514756657

Offset: 1

Author

Hugo Pfoertner, Feb 02 2020

Keywords

• nonn

Comments

Comparison of actual and approximated number of primes < 10^n:

Limit

10^n

| A083844(n)

| | a(n)

| | | (a(n) - A083844(n))/A083844(n)

10^1 2 1 -0.50000

10^2 4 4 0.0

10^3 10 9 -0.10000

10^4 19 20 0.052632

10^5 51 48 -0.058824

10^6 112 121 0.080357

10^7 316 317 0.0031646

10^8 841 855 0.016647

10^9 2378 2356 -0.0092515

10^10 6656 6609 -0.0070613

10^11 18822 18787 -0.0018595

10^12 54110 53970 -0.0025873

10^13 156081 156385 0.0019477

10^14 456362 456404 9.2032E-5

10^15 1339875 1340088 0.00015897

10^16 3954181 3955219 0.00026251

10^17 11726896 11726332 -4.8095E-5

10^18 34900213 34903256 8.7191E-5

10^19 104248948 104251560 2.5055E-5

10^20 312357934 312353236 -1.5040E-5

10^21 938457801 938461459 3.8979E-6

10^22 2826683630 2826668497 -5.3536E-6

10^23 8533327397 8533343468 1.8833E-6

10^24 25814570672 25814350227 -8.5396E-6

10^25 78239402726 78239112814 -3.7054E-6

10^26 237542444180 237541788793 -2.7590E-6

10^27 722354138859 722354115787 -3.1940E-8

10^28 2199894223892 2199893807666 -1.8920E-7

Links

• Keith Conrad, Hardy-Littlewood Constants, (2003).

Crossrefs

Cf. A083844, A331941.

Programs

PARI

C=0.68640673140912300455609634836350943408916655062787977896811707366392;
x=1.0;S10=sqrt(10);for(k=1,30,x*=s10;print1(round(C*intnum(y=2,x,1/log(y))),", "))

Formula

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)).