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.

Showing 1-3 of 3 results.

A206815 Position of n+pi(n) in the joint ranking of {j+pi(j)} and {k+(k+1)/log(k+1)}.

Original entry on oeis.org

1, 2, 5, 7, 9, 11, 14, 15, 17, 19, 22, 23, 26, 28, 30, 31, 34, 36, 39, 40, 42, 44, 47, 49, 50, 52, 54, 56, 58, 60, 63, 65, 67, 68, 70, 72, 75, 77, 78, 80, 83, 85, 87, 89, 91, 93, 96, 98, 99, 101, 103, 105, 108, 109, 111, 113, 115, 117, 119, 121, 124, 126, 128
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2012

Keywords

Comments

The sequences A206815, A206818, A206827, A206828 illustrate the closeness of {j+pi(j)} to {k+(k+1)/log(k+1)}, as suggested by the prime number theorem and the conjecture that all the terms of A206827 and A206828 are in the set {1,2,3}.

Examples

			The joint ranking is represented by
1 < 3 < 3.8 < 4.7 < 5 < 5.8 < 6 <7.1 < 8 < 8.3 < 9 < ...
Positions of numbers j+pi(j): 1,2,5,7,9,...
Positions of numbers k+(k+1)/log(k+1): 3,4,6,8,10,..

Crossrefs

Cf. A000720, A206827, A206818 (complement of A206815).

Programs

  • Mathematica
    f[1, n_] := n + PrimePi[n];
f[2, n_] := n + N[(n + 1)/Log[n + 1]]; z = 500;
t[k_] := Table[f[k, n], {n, 1, z}];
t = Sort[Union[t[1], t[2]]];
p[k_, n_] := Position[t, f[k, n]];
Flatten[Table[p[1, n], {n, 1, z}]]    (* A206815 *)
Flatten[Table[p[2, n], {n, 1, z}]]    (* A206818 *)
d1[n_] := p[1, n + 1] - p[1, n]
Flatten[Table[d1[n], {n, 1, z - 1}]]  (* A206827 *)
d2[n_] := p[2, n + 1] - p[2, n]
Flatten[Table[d2[n], {n, 1, z - 1}]]  (* A206828 *)

A206588 Number of solutions k of prime(k)=prime(n) (mod n), where 1<=k

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 2, 0, 2, 1, 2, 1, 1, 1, 2, 1, 1, 0, 2, 0, 3, 1, 2, 2, 3, 1, 3, 1, 1, 2, 2, 1, 3, 1, 3, 2, 2, 1, 3, 1, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 3, 0, 3, 0, 1, 1, 2, 0, 4, 1, 2, 1, 3, 1, 5, 1, 1, 0, 1, 0, 2, 0, 2, 1, 2
Offset: 2

Views

Author

Clark Kimberling, Feb 09 2012

Keywords

Comments

In the following guide to related sequences, c(n) is the number of solutions (n,k) of s(k)=s(n) (mod n), where 1<=k
s(n).............c(n)
prime(n).........A206588
prime(n+1).......A206589
n^2..............A057918
n^3..............A206590
Fibonacci(n+1)...A206713
2^(n-1)..........A206714
n!...............A072480
n(n+1)/2.........A206824
n^4..............A206825
n(n+1)(n+2)/6....A206826
n(n+1)(2n+1)/6...A206827
C(2n,n)..........A206828
For some choices of s, the limiting frequency of 0's in c appears to be a positive constant.

Examples

			For k=1 to 7, the numbers p(8)-p(k) are 17,16,14,12,8,6,4, so that a(8)=2.

Crossrefs

Cf. A206589.

Programs

  • Mathematica
    f[n_, k_] := If[Mod[Prime[n] - Prime[k], n] == 0, 1, 0];
t[n_] := Flatten[Table[f[n, k], {k, 1, n - 1}]]
a[n_] := Count[Flatten[t[n]], 1]
Table[a[n], {n, 2, 120}]  (* A206588 *)

A206818 Position of n+(n+1)/log(n+1) in the joint ranking of {j+pi(j)} and {k+(k+1)/log(k+1)}.

Original entry on oeis.org

3, 4, 6, 8, 10, 12, 13, 16, 18, 20, 21, 24, 25, 27, 29, 32, 33, 35, 37, 38, 41, 43, 45, 46, 48, 51, 53, 55, 57, 59, 61, 62, 64, 66, 69, 71, 73, 74, 76, 79, 81, 82, 84, 86, 88, 90, 92, 94, 95, 97, 100, 102, 104, 106, 107, 110, 112, 114, 116, 118, 120, 122, 123
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2012

Keywords

Comments

The sequences A206815, A206818, A206827, A206828 illustrate the closeness of {j+pi(j)} to {k+(k+1)/log(k+1)}, as suggested by the prime number theorem and the conjecture that all the terms of A206827 and A206828 are in the set {1,2,3}.

Examples

			The joint ranking is represented by
1 < 3 < 3.8 < 4.7 < 5 < 5.8 < 6 <7.1 < 8 < 8.3 < 9 < ...
Positions of numbers j+pi(j): 1,2,5,7,9,...
Positions of numbers k+(k+1)/log(k+1): 3,4,6,8,10,..

Crossrefs

Cf. A000720, A206827, A206815 (complement of A206818).

Programs

  • Mathematica
    f[1, n_] := n + PrimePi[n];
f[2, n_] := n + N[(n + 1)/Log[n + 1]]; z = 500;
t[k_] := Table[f[k, n], {n, 1, z}];
t = Sort[Union[t[1], t[2]]];
p[k_, n_] := Position[t, f[k, n]];
Flatten[Table[p[1, n], {n, 1, z}]]    (* A206815 *)
Flatten[Table[p[2, n], {n, 1, z}]]    (* A206818 *)
d1[n_] := p[1, n + 1] - p[1, n]
Flatten[Table[d1[n], {n, 1, z - 1}]]  (* A206827 *)
d2[n_] := p[2, n + 1] - p[2, n]
Flatten[Table[d2[n], {n, 1, z - 1}]]  (* A206828 *)
Showing 1-3 of 3 results.

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