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A003066 Problimes (first definition).
(Formerly M0997)

%I M0997

%S 2,4,6,9,12,15,19,23,27,31,35,40,45,50,55,60,65,70,75,80,86,92,98,104,

%T 110,116,122,128,134,140,146,152,158,164,170,176,182,188,194,200,206,

%U 213,220,227,234,241,248,255,262,269,276,283,290,297,304,311,318,325

%N Problimes (first definition).

%C From _Dean Hickerson_, Jan 13 2003: (Start)

%C 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.

%C 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. (End)

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A003066/b003066.txt">Table of n, a(n) for n = 1..1000</a>

%H M. D. Hirschhorn, <a href="http://www.jstor.org/stable/2319173">How unexpected is the prime number theorem?</a>, Amer. Math. Monthly, 80 (1973), 675-677.

%H M. D. Hirschhorn, <a href="/A003066/a003066.pdf">How unexpected is the prime number theorem?</a>, Amer. Math. Monthly, 80 (1973), 675-677. [Annotated scanned copy]

%H R. C. Vaughan, <a href="http://blms.oxfordjournals.org/content/6/3/337.extract">The problime number theorem</a>, Bull. London Math. Soc., 6 (1974), 337-340.

%p 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: # _James A. Sellers_, Mar 07 2000

%t a[1] = 2; a[n_] := a[n] = Floor[a[n-1] + 1/Product[1-1/a[j], {j, 1, n-1}]]; Table[a[n], {n, 1, 60}] (* _Jean-Fran├žois Alcover_, Mar 09 2012, after _James A. Sellers_ *)

%Y Cf. A003067, A003068.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Mar 07 2000

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Last modified July 25 13:05 EDT 2021. Contains 346290 sequences. (Running on oeis4.)