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A099260
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Number of decimal digits in (10^n)-th prime number.
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2
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1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75
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OFFSET
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0,2
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COMMENTS
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As lim {n->oo} p_n/(n log n) = 1 is equivalent to the prime number theorem, a good first approximation (without having done any detailed analysis) should be a(n)=floor(log_10((10^n)*log(10^n))), which correctly generates all the first 22 terms and predicts that the sequence will continue 24,25,...,43,44,46,47,...,435,436,438,439,...,4344,4345,4347,4348,...,4503,4504 through the first 4500 terms (with only 5,45,437,4346 not appearing - compare with the digits of log_10(e) in A002285).
Many terms of this sequence can be determined exactly using Dusart's bounds. The first missing terms are 5, 44, 435, 4344, 43430, 434295, 4342946, 43429449, 434294483, 4342944820, ....
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LINKS
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EXAMPLE
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a(4) = 6 because A006988(4) = prime(10^4) = 104729 has six decimal digits.
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MATHEMATICA
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Table[IntegerLength[Prime[10^n]], {n, 0, 75}] (* Harvey P. Dale, Dec 11 2020 *)
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PROG
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(PARI) a(n)=if(n<3, return(n+1)); my(l=n*log(10), ll=log(l), lb=ceil(log(l+ll-1+(ll-2.2)/l)/log(10)), ub=ceil(log(l+ll-1+(ll-2)/l)/log(10))); if(lb==ub, n+lb, error("Cannot determine a("n")"))
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CROSSREFS
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KEYWORD
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nonn,base,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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