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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). 2
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 (list; graph; refs; listen; history; text; internal format)
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, remark first that the square root 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 has always 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: A128419 A226945 A225137 * A006880 A227693 A175255

Adjacent sequences:  A229252 A229253 A229254 * A229256 A229257 A229258

KEYWORD

nonn,less

AUTHOR

Vladimir Pletser, Sep 17 2013

STATUS

approved

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Last modified April 19 08:44 EDT 2019. Contains 322241 sequences. (Running on oeis4.)