A229255 Integer nearest to (2^(n-1) + 3^(n-1))^(2*b(n)) where b(n) = (C1*n^(Pi)*exp(C2*n)*cos(C3*n+C4) + C5)*(C6*n^C7 + Pi/2) (see coefficients in comments).

4, 25, 168, 1229, 9592, 78488, 664356, 5761311, 50857532, 455110791, 4117706679, 37598394076, 345973354409, 3204537723387, 29847287869987, 279317953220125, 2624541016148480, 24747919106286414, 234089443816438414, 2220530456953251916, 21119025631088169139, 201358809736398135352, 1924434871799161020533, 18434884359943473267194, 176994218822287711757127

Offset

1,1

Comments

Coefficients are C1=27829/125000000, C2=-0.591561, C3=441/2500, C4=5, C5=19703973/31250000, C6=5.241804273*10^-3, C7=0.6246728093.

This sequence gives a good approximation of pi(10^n) (A006880); see (A229256).

To obtain this sequence, note first that the square roots of the first values of pi(10^n) (A006880) (see A221205) are equal or close to some values of A229194, i.e., A221205(n) = or ≈ A229194(2n+1) = round(2^(n-1) + 3^(n-1)) for 1 <= n <= 25. Then, values of pi(10^n), A006880(n) = or ≈ (A229194(2n+1))^2 = round((2^(n-1) + 3^(n-1)))^2 for 1 <= n <= 25. Finally, the fit is improved by multiplying the exponent 2 by the sequence b(n) which always has values close to one for 1 <= n <= 25, varying between 0.99382... and 1.01511....

Links

Vladimir Pletser, Table of n, a(n) for n = 1..500

Eric Weisstein's World of Mathematics, Prime Counting Function

Formula

a(n) = round((2^(n-1) + 3^(n-1))^(2*(C1*n^(Pi)*exp(C2*n)*cos(C3*n+C4) + C5)*(C6*n^C7 + Pi/2))).

Example

For n=1, b(1) = (C1*exp(C2)*cos(C3+C4) + C5)*(C6 + Pi/2))= 0.99382..., then a(1) = round(2^(2*0.99382...)) = round(3.96587...) = 4.

Maple

C1:=27829/125000000: C2:=-5.91561e-01: C3:=441/2500: C4:=5: C5:=19703973/31250000: C6:=5.241804273e-03: C7:=6.246728093e-01: b:=n-> (C1*n^(Pi)*exp(C2*n)*cos(C3*n+C4)+C5)*(C6*n^C7+(Pi/2)):  seq(round((2^(n-1)+3^(n-1))^(2*b(n))), n=1..25);

Crossrefs

Cf. A006880, A221205, A229194, A229256.

Sequence in context: A372216 A226945 A225137 * A006880 A227693 A175255

Adjacent sequences: A229252 A229253 A229254 * A229256 A229257 A229258

Keyword

nonn,less

Author

Vladimir Pletser, Sep 17 2013

Status

approved