This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A003066 M0997 #34 Jul 02 2025 16:01:54
%S A003066 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 A003066 110,116,122,128,134,140,146,152,158,164,170,176,182,188,194,200,206,
%U A003066 213,220,227,234,241,248,255,262,269,276,283,290,297,304,311,318,325
%N A003066 Problimes (first definition).
%C A003066 From _Dean Hickerson_, Jan 13 2003: (Start)
%C A003066 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 A003066 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 A003066 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003066 T. D. Noe, <a href="/A003066/b003066.txt">Table of n, a(n) for n = 1..1000</a>
%H A003066 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 A003066 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 A003066 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 A003066 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 Sellers_, Mar 07 2000
%t A003066 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-François Alcover_, Mar 09 2012, after _James Sellers_ *)
%Y A003066 Cf. A003067, A003068.
%K A003066 nonn,nice
%O A003066 1,1
%A A003066 _N. J. A. Sloane_
%E A003066 More terms from _James Sellers_, Mar 07 2000