A293696 -id:A293696 - cp's OEIS frontend

A293697 a(n) is the sum of prime numbers between 2^n+1 and 2^(n+1).

2, 3, 12, 24, 119, 341, 1219, 4361, 16467, 57641, 208987, 780915, 2838550, 10676000, 39472122, 148231324, 559305605, 2106222351, 7995067942, 30299372141, 115430379568, 440354051430, 1683364991290, 6448757014608, 24754017328490, 95132828618112, 366232755206338

Offset: 0

Olivier Gérard, Oct 15 2017

nonn

			From _Gus Wiseman_, Jun 02 2024: (Start)
Row-sums of:
   2
   3
   5   7
  11  13
  17  19  23  29  31
  37  41  43  47  53  59  61
  67  71  73  79  83  89  97 101 103 107 109 113 127
(End)

Cf. A036378 (number of primes summed).
Cf. A293696 (triangle of partial sums).
Minimum is A014210 or A104080, indices A372684.
Maximum is A014234, delta A013603.
Counting all numbers (not just prime) gives A049775.
For squarefree instead of prime numbers we have A373123, length A077643.
For prime indices we have A373124.
Partial sums give A130739(n+1).
Cf. A000040, A001223, A029931, A035100, A046933, A061398, A092131.

Table[Plus @@
  Table[Prime[i], {i, PrimePi[2^(n)] + 1, PrimePi[2^(n + 1)]}], {n, 0,
   24}]

A293677 a(n) is the greatest integer k such that k/Fibonacci(n) < Pi.

Original entry on oeis.org

0, 3, 3, 6, 9, 15, 25, 40, 65, 106, 172, 279, 452, 731, 1184, 1916, 3100, 5017, 8117, 13134, 21252, 34387, 55640, 90028, 145669, 235697, 381367, 617065, 998432, 1615498, 2613930, 4229428, 6843359, 11072788, 17916147, 28988936, 46905084, 75894020, 122799104

Offset: 0

Clark Kimberling, Oct 16 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000045, A000796, A293678, A293679.

z = 120; f[n_] := Fibonacci[n];
Table[Floor[Pi*f[n]], {n, 0, z}];   (* A293677 *)
Table[Ceiling[Pi*f[n]], {n, 0, z}]; (* A293678 *)
Table[Round[Pi*f[n]], {n, 0, z}];   (* A293696 *)

a(n) = floor(Pi*Fibonacci(n)).

a(n) = A293678(n) - 1 for n > 0.

A294154 Base-2 logarithm of product of primes between 2^n+1 and 2^(n+1).

Original entry on oeis.org

1, 1, 5, 7, 22, 39, 84, 173, 367, 716, 1444, 2945, 5823, 11817, 23457, 47117, 94496, 188670, 378296, 755437, 1512435, 3024742, 6049260, 12099777, 24205776, 48402649, 96809896, 193632363, 387252738, 774544273, 1549044943

Offset: 0

Olivier Gérard, Oct 24 2017

Inspired by Bertrand's postulate and complexity of algorithms using Farey fraction sets of size 2^n.

The log of this sequence seems linear after a few terms.

Cf. A036378 (number of primes between 2^n+1 and 2^(n+1)).

Cf. A293696, A293697 (additive view of the primes between 2^n+1 and 2^(n+1)).

Table[Floor[
  Log[2, Times @@
    Table[Prime[i], {i, PrimePi[2^(n)] + 1,
      PrimePi[2^(n + 1)]}]]], {n, 0, 20}]

a(26)-a(30) from Robert Price, Nov 30 2017

cp's OEIS Frontend

A293697 a(n) is the sum of prime numbers between 2^n+1 and 2^(n+1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A293677 a(n) is the greatest integer k such that k/Fibonacci(n) < Pi.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A294154 Base-2 logarithm of product of primes between 2^n+1 and 2^(n+1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions