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Search: id:A003066
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| A003066 |
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Problimes (first definition).
(Formerly M0997)
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+0
3
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| 2, 4, 6, 9, 12, 15, 19, 23, 27, 31, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 213, 220, 227, 234, 241, 248, 255, 262, 269, 276, 283, 290, 297, 304, 311, 318, 325
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Comments from Dean Hickerson, Jan 13, 2003: Suppose you have a list of the first n prime numbers p_1, ..., p_n and you want to estimate the next one. The probability that a random integer is not divisible by any of p_1, ..., p_n is (1-1/p_1) * ... * (1-1/p_n). In other words, 1 out of every 1/((1-1/p_1) * ... * (1-1/p_n)) integers is relatively prime to p_1, ..., p_n.
So we might expect the next prime to be roughly this much larger than p_n; i.e. p_(n+1) may be about p_n + 1/((1-1/p_1) * ... * (1-1/p_n)). This sequence and A003067, A003068 are obtained by replacing this approximation by an exact equation, using 3 different ways of making the results integers.
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REFERENCES
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M. D. Hirschhorn, How unexpected is the prime number theorem?, Amer. Math. Monthly, 80 (1973), 675-677.
R. C. Vaughan, The problime number theorem, Bull. London Math. Soc., 6 (1974), 337-340.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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MAPLE
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a[1] := 2: for i from 1 to 150 do a[i+1] := floor(a[i]+1/product((1-1/a[j]), j=1..i)): od:
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CROSSREFS
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Cf. A003067, A003068.
Adjacent sequences: A003063 A003064 A003065 this_sequence A003067 A003068 A003069
Sequence in context: A143118 A083652 A118103 this_sequence A075349 A130025 A076271
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KEYWORD
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nonn,nice
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AUTHOR
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njas
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EXTENSIONS
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More terms and Maple code from James A. Sellers (sellersj(AT)math.psu.edu), Mar 07 2000
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