A003066 Problimes (first definition). (Formerly M0997)

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

Offset

1,1

Comments

From Dean Hickerson, Jan 13 2003: (Start)

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. (End)

References

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

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

M. D. Hirschhorn, How unexpected is the prime number theorem?, Amer. Math. Monthly, 80 (1973), 675-677.

M. D. Hirschhorn, How unexpected is the prime number theorem?, Amer. Math. Monthly, 80 (1973), 675-677. [Annotated scanned copy]

R. C. Vaughan, The problime number theorem, Bull. London Math. Soc., 6 (1974), 337-340.

Maple

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

Mathematica

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 A. Sellers *)

Crossrefs

Cf. A003067, A003068.

Sequence in context: A185601 A157795 A294060 * A075349 A156024 A234363

Adjacent sequences: A003063 A003064 A003065 * A003067 A003068 A003069

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Mar 07 2000

Status

approved