A003068 -id:A003068 - cp's OEIS frontend

A178532 Partial sums of problimes (third definition, A003068).

2, 6, 13, 24, 39, 58, 81, 109, 142, 180, 223, 271, 324, 382, 445, 513, 586, 665, 750, 841, 938, 1041, 1150, 1265, 1386, 1513, 1646, 1785, 1930, 2081, 2238, 2401, 2570, 2745, 2926, 3113, 3306, 3505, 3710, 3921, 4138, 4362, 4593, 4831, 5076

Offset: 1

Jonathan Vos Post, Dec 28 2010

The subsequence of prime partial sums of problimes begins: 2, 13, 109, 223, 271, 2081, 4831, 8233.

The subsequence of problime partial sums of problimes begins: 2, 58, 109.

			a(12) = 2 + 4 + 7 + 11 + 15 + 19 + 23 + 28 + 33 + 38 + 43 + 48 = 271 is prime.

Cf. A003068, A007504, A013918.

SUM[i=1..n] A003068(i).

A003066 Problimes (first definition).

Original entry on oeis.org

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

N. J. A. Sloane

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)

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. A003067, A003068.

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

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 *)

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

A062467 Integer part of log(n!)^(1 + log(n)/n).

Original entry on oeis.org

0, 0, 2, 4, 7, 11, 15, 19, 23, 28, 32, 37, 41, 46, 50, 55, 60, 64, 69, 74, 78, 83, 88, 93, 97, 102, 107, 112, 116, 121, 126, 131, 136, 141, 145, 150, 155, 160, 165, 170, 175, 180, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 239, 244, 249, 254, 259

Offset: 1

Olivier Gérard, Jun 23 2001

Cf. A062468, see also A003068.

Mathematica
```
Floor[Log[n! ]^(1 + Log[n]/n)]
```

a(n) = floor(log(n!)^(1 + log(n)/n)). - Wesley Ivan Hurt, Mar 15 2023

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

A178907 n-th prime + n-th problime (third definition).

Original entry on oeis.org

4, 7, 12, 18, 26, 32, 40, 47, 56, 67, 74, 85, 94, 101, 110, 121, 132, 140, 152, 162, 170, 182, 192, 204, 218, 228, 236, 246, 254, 264, 284, 294, 306, 314, 330, 338, 350, 362, 372, 384, 396, 405, 422, 431, 442, 451, 470, 489, 500, 509, 520, 533, 542, 559, 572, 585, 598, 607, 620, 631, 640, 657, 678

Offset: 1

Jonathan Vos Post, Dec 29 2010

nonn

The partial sequence of values which are themselves primes begins: 7, 47, 101. The partial sequence of values which are themselves problimes (third definition) begins: 4, 7, 85, 121.

			a(10) = prime(10) + A003068(10) = 29 + 38 = 67 is prime.

Cf. A000040, A003068, A100484.

a(n) = A000040(n) + A003068(n).

Corrected and extended. - R. J. Mathar, Oct 20 2013

cp's OEIS Frontend

A178532 Partial sums of problimes (third definition, A003068).

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A003066 Problimes (first definition).

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A003067 Problimes (second definition).

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A062467 Integer part of log(n!)^(1 + log(n)/n).

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

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A178907 n-th prime + n-th problime (third definition).

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