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A141306 Approximate prime counts at 10^n resulting from an experimental algorithm designed to estimate with 99+% accuracy unknown values of 10^n. 2
22, 166, 1107, 9222, 73778, 645555, 5533333, 49799993, 442666605, 4057777214, 36888883762, 342539634932, 3161904322448, 29642853022951, 276666628214208, 2612962599800854, 24592589174596271, 233629597158664575, 2213333025713664398 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

The purpose of this sequence is to encourage experimentation. The number of primes at each power of 10 through 10^23 is now known exactly. Using this information, I created, by trial and error, an algorithm to check its accuracy against known counts (see UBASIC program) and to predict rough approximate values for those counts yet to be discovered. At increasing powers of 10 this program approaches 100% accuracy.

In the algorithm below, the value 9.14615 may be changed to 10 or to any other value between 9.1 and 10. With value 9.14615 the predicted value at 10^21 is ~100% accurate and with value 9.1151 the predicted value at 10^23 is also ~100% accurate and at some point the correct value may converge on a value of 9.1.

Starting at 10^2, the percent of accuracy is less than the next at 10^3 and the values undulate alternately between <, >, <, >, . . . thereafter.

Using 9.14615 (accurate for a limit of 10^21) in the program and beginning at 10^2, the estimate compared to the true value - the percent accuracy (truncated, not rounded) is: 10^2 - 87.9%, 10^3 - 98.8%, 10^4 - 90.0%, 10^5 - 96.1%, 10^6 - 93.9%, 10^7 - 97.1%, 10^8 - 96.0%, 10^9 - 97.9%, 10^10 - 97.2%, 10^11 - 98.5%, 10^12 - 98.0%, 10^13 - 98.9%, 10^14 - 98.6%, 10^15 - 99.3, 10^16 - 99.0%, 10^17 - 99.5%, 10^18 - 99.4%, 10^19 - 99.8%, 10^20 - 99.6%, 10^21 - ~100.0%

Using 9.1151 (accurate for a limit of 10^23) in the program, at 10^22 - 99.7% and 10^23 - 99.9% or ~100%.

pi(10^n) ~ 10^n / log(10^n). The divisor Y in this sequence is floor(n^2/2), so the sequence varies as 10^n/n * 0.442666605... while pi(10^n) varies as 10^n/n * 0.434294481....

REFERENCES

Perri O'Shaughnessy, Case of Lies. NY, Dell, 2006. A rare legal mystery incorporating some fascinating reading on encryption, prime numbers and related topics specifically relevant to this sequence (see Epilogue).

David Wells, Prime Numbers, The Most Mysterious Figures in Math. NJ, Wiley, 2005. Pages 183-185. At the time, 10^21 was the limit. See especially page 185, estimating p(n).

LINKS

Table of n, a(n) for n=2..20.

C. Caldwell's Prime Pages, A Table of values of pi(x).

Enoch Haga, UBASIC Program

FORMULA

N=(log(9.14615^X)*10^(X-1))/Y where X=power of 10 and Y=divisor beginning at 2 and thereafter by differencing two successive terms so that we have y = 0, 2, 4, 8, 12, 18, 24 with differences 2,2,4,4,6,6,8,8, etc. Above a(1)=0 because at 10^1 x=1 and y=0 so is omitted.

a(n) = round(log(9.14615^n)*10^(n-1)/floor(n^2/2))

a(n) ~ 0.442666605... * 10^n/n

EXAMPLE

a(4)=1107 because when X=4 and Y=8, then N=(log(9.14615^X)*10^(X-1))/Y = 1107.

PROG

(PARI) a(n)=round(log(9.14615^n)*10^(n-1)/(n^2\2))

CROSSREFS

Cf. A006880.

Sequence in context: A224185 A185859 A173393 * A067561 A223768 A224156

Adjacent sequences:  A141303 A141304 A141305 * A141307 A141308 A141309

KEYWORD

easy,nonn

AUTHOR

Enoch Haga, Jun 25 2008

EXTENSIONS

Correction, comment, program, formulas, and editing from Charles R Greathouse IV, Nov 02 2009

STATUS

approved

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Last modified October 15 21:38 EDT 2021. Contains 348034 sequences. (Running on oeis4.)