A003066 -id:A003066 - cp's OEIS frontend

A003068 Problimes (third definition).

2, 4, 7, 11, 15, 19, 23, 28, 33, 38, 43, 48, 53, 58, 63, 68, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329

Offset: 1

N. J. A. Sloane

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

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.

Cf. A003066, A003067.

a[1] := 2: for i from 1 to 150 do a[i+1] := ceil(a[i]+1/product((1-1/a[j]), j=1..i)): od: # James Sellers, Mar 07 2000

a[1] = 2; a[n_] := a[n] = Ceiling[a[n-1] + 1/Product[1 - 1/a[j], {j, 1, n-1}]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Nov 18 2013 *)

More terms from James Sellers, Mar 07 2000

A003067 Problimes (second definition).

Original entry on oeis.org

2, 4, 7, 10, 13, 17, 21, 25, 29, 34, 39, 44, 49, 54, 59, 64, 69, 74, 79, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 235, 242, 249, 256, 263, 270, 277, 284, 291, 298, 305, 312, 319

Offset: 1

N. J. A. Sloane

It would be nice to have a clearer definition. - N. J. A. Sloane, Jul 21 2008

The g.f. (z**2+2+z**9+z**5)/(z-1)**2 conjectured by Simon Plouffe in his 1992 dissertation is wrong.

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

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.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Cf. A003066, A003068.

a[1] := 2: for i from 1 to 150 do a[i+1] := round(a[i]+1/product((1-1/a[j]), j=1..i)): od: # James Sellers, Mar 07 2000

a[1] = 2; a[n_] := a[n] = Round[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 *)

More terms from James Sellers, Mar 07 2000

A178899 Numbers which are both primes and problimes (third definition).

Original entry on oeis.org

2, 7, 11, 19, 23, 43, 53, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 1873, 1999, 2017, 2053, 2089, 2143, 2161, 2179, 2251, 2269, 2287, 2341, 2377, 2467, 2503, 2521, 2539, 2557, 2593, 2647, 2683, 2719, 2791, 2917, 2953, 2971, 3061, 3079

Offset: 1

Jonathan Vos Post, Dec 29 2010

nonn

Alois P. Heinz, 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.

Cf. A000040, A003068, A178532.

b:= proc(n) option remember; local k;
      if n=1 then c(2):= 1; 2
             else k:= ceil(b(n-1) +1/mul((1-1/b(j)), j=1..n-1));
                  c(k):= n; k
      fi
    end:
a:= proc(n) option remember; local k;
      if n=1 then b(1)
             else for k from c(a(n-1))+1 while not isprime(b(k))
                  do od; b(k)
      fi
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 29 2010

nmax = 400;
b[n_] := b[n] = If[n==1, 2, Ceiling[b[n-1]+1/Product[1-1/b[j], {j, 1, n-1}]]];
Intersection[Array[b, nmax], Prime[Range[PrimePi[b[nmax]]]]] (* Jean-François Alcover, Nov 20 2020 *)

A000040 INTERSECTION A003068.

More terms from Alois P. Heinz, Dec 29 2010

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A003068 Problimes (third definition).

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A003067 Problimes (second definition).

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A178899 Numbers which are both primes and problimes (third definition).

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