cp's OEIS Frontend

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.

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A250622 a(n) = floor(n*log(prime(n)))-prime(n), n >= 1.

Original entry on oeis.org

-2, -1, -1, 0, 0, 2, 2, 4, 5, 4, 6, 6, 7, 9, 10, 10, 10, 12, 12, 14, 17, 17, 18, 18, 17, 18, 22, 23, 27, 28, 23, 25, 25, 28, 26, 29, 30, 30, 32, 33, 33, 37, 34, 38, 40, 44, 40, 36, 38, 42, 45, 45, 49, 47, 48, 49, 49, 53, 54, 57, 61, 59, 53, 56, 60, 63, 57, 58, 56, 60, 63, 64, 64, 65, 66, 69, 70, 69
Offset: 1

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Author

Freimut Marschner, Dec 02 2014

Keywords

Comments

Since n*log(prime(n)) > prime(n), n >= 4 and ceiling(prime(n) - n*log(n)) < prime(n), then n*log(n) < prime(n) < n*log(prime(n)), n >= 4. This inequality is included in the prime number theorem PNT. Remark: a(n) >= 0 for n >=4 otherwise a(n) < 0.

Examples

			n = 1, a(1) = floor(1*0.6931...) - 2 = 0 - 2 = -2;
n = 5, a(5) = floor(5*2.3978...) - 11 = floor( 11.9894...) - 11 = 11 - 11 = 0;
n = 6, a(6) = floor(6*2.5649...) - 13 = floor(15.3896...) - 13 = 15 - 13 = 2.

Links

Crossrefs

Cf. A000040, A064658 (ceiling(prime(n) - n*log(n))), A250621 (floor(n*log(prime(n)))).

Programs

  • Mathematica
    a250622[n_Integer] := Table[Floor[i*Log[Prime[i]]] - Prime[i], {i, n}]; a250622[121] (* Michael De Vlieger, Dec 11 2014 *)
  • PARI
    vector(100,n,floor(n*log(prime(n))-prime(n))) \\ Derek Orr, Dec 13 2014

Formula

a(n) = floor(n*log(prime(n))) - prime(n) = A250621(n) - A000040(n).

A250623 a(n) = floor(n*log(prime(n))) + ceiling(n*log(n)) - 2*prime(n).

Original entry on oeis.org

-4, -2, -2, -1, -2, 0, -1, 2, 2, -1, 2, -1, 0, 3, 4, 2, 0, 4, 1, 3, 8, 7, 8, 6, 1, 2, 8, 10, 16, 18, 3, 5, 4, 9, 2, 8, 7, 6, 8, 8, 7, 13, 5, 12, 15, 22, 10, -1, 2, 9, 13, 12, 19, 12, 12, 12, 11, 18, 18, 22, 29, 22, 8, 12, 19, 23, 8, 8, 2, 9, 13, 13, 11, 11, 11
Offset: 1

Views

Author

Freimut Marschner, Dec 02 2014

Keywords

Comments

It is known that n*log(n) < prime(n) < n*prime(n), n >= 4. The arithmetic mean of the limits of this inequality is f(n) = (floor((n*log(n)) + ceiling(n*prime(n))))/2. So a(n) is the difference between twice this quantity and 2*prime(n).

Examples

			a(4) = floor(4*log(7)) + ceiling(4*log(4)) - 2*7 = floor(7.78...) + ceiling(5.54...) - 14 = 7 + 6 - 14 = -1;
a(6) = floor(6*log(13)) + ceiling(6*log(6)) - 2*13 = floor(15.38...) + ceiling(10.75..) - 26 = 15 + 11 - 26 = 0.

Links

Programs

  • Mathematica
    a250623[n_] :=
Floor[#*Log[Prime[#]]] + Ceiling[#*Log[#]] - 2*Prime[#] & /@ Range[n]; a250623[137] (* Michael De Vlieger, Dec 26 2014 *)
  • PARI
    vector(100,n,floor(n*log(prime(n)))+ceil(n*log(n))-2*prime(n)) \\ Derek Orr, Dec 30 2014

Formula

a(n) = A250621(n) + A050502(n) - 2*A000040(n).
Showing 1-2 of 2 results.

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