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 1-4 of 4 results.

A358619 First forward difference of A258037.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Clark Kimberling and Robert G. Wilson v, Oct 31 2022

Keywords

Comments

Conjecture: All terms belong to {1, 2, 3}. See third comment in A258037.

Crossrefs

Cf. A258037, A358618.

Programs

  • Mathematica
    nn = 210; p = Prime@ Range@ nn; t = Table[ Differences[p, n][[1]], {n, 0, nn - 1}]; s = Select[ Range@ nn, t[[#]] > 0 &]; d = Differences@ s

A258036 Numbers k such that D(prime(k), k-1) < 0, where D( * , k-1) = (k-1)-st difference.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 17, 19, 21, 23, 25, 28, 30, 32, 34, 36, 38, 41, 43, 45, 47, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 101, 103, 105, 107, 109, 111, 114, 116, 118, 120, 122, 124, 126, 128
Offset: 1

Views

Author

Clark Kimberling, Jun 05 2015

Keywords

Comments

Partition of the positive integers: A258036, A258037;
Corresponding partition of the primes: A258038, A258039.
Do all the terms of the difference sequence of A258036 belong to {1,2,3}?

Examples

			D(prime(2), 1) = 3 - 2 > 0;
D(prime(3), 2) = 5 - 2*3 + 2 > 0;
D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0, so a(1) = 4;

Links

Crossrefs

Cf. A258037, A258038, A258039.

Programs

  • Mathematica
    u = Table[Prime[Range[k]], {k, 1, 1000}];
v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];
w1 = Flatten[Position[v, -1]] (* A258036 *)
w2 = Flatten[Position[v, 1]]  (* A258037 *)
Prime[w1]  (* A258038 *)
Prime[w2]  (* A258039 *)
  • PARI
    is(k) = {my(p=primes(k));sum(i=0,k-1,(-1)^i*p[k-i]*binomial(k-1,i))<0} \\ Jason Yuen, Nov 13 2024

Formula

D(prime(k), k-1) = Sum_{i=0..k-1} (-1)^i*prime(k-i)*binomial(k-1,i). [corrected by Jason Yuen, Nov 13 2024]

A258038 Numbers prime(k) such that D(prime(k), k-1) < 0, where D( * , k-1) = (k-1)-st difference.

Original entry on oeis.org

7, 13, 19, 29, 37, 43, 59, 67, 73, 83, 97, 107, 113, 131, 139, 151, 163, 179, 191, 197, 211, 223, 229, 239, 251, 263, 271, 281, 293, 311, 317, 337, 349, 359, 373, 383, 397, 409, 421, 433, 443, 457, 463, 479, 491, 503, 521, 523, 547, 563, 571, 587, 599, 607
Offset: 1

Views

Author

Clark Kimberling, Jun 05 2015

Keywords

Comments

Partition of the positive integers: A258036, A258037;
Corresponding partition of the primes: A258038, A258039.

Examples

			D(prime(2), 1) = 3 - 2 > 0;
D(prime(3), 2) = 5 - 2*3 + 2 > 0;
D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0, so a(1) = prime(4) = 7;

Links

Crossrefs

Cf. A258036, A258037, A258039.

Programs

  • Mathematica
    u = Table[Prime[Range[k]], {k, 1, 1000}];
v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];
w1 = Flatten[Position[v, -1]] (* A258036 *)
w2 = Flatten[Position[v, 1]]  (* A258037 *)
p1 = Prime[w1]  (* A258038 *)
p2 = Prime[w2]  (* A258039 *)

Formula

D(prime(k), k-1) = Sum_{i=0..k-1} (-1)^i*prime(k-i)*binomial(k-1,i). [corrected by Jason Yuen, Nov 13 2024]
a(n) = prime(A258036(n)). - Jason Yuen, Nov 13 2024

A258039 Numbers prime(k) such that D(prime(k), k-1) > 0, where D( * , k-1) = (k-1)-st difference.

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 31, 41, 47, 53, 61, 71, 79, 89, 101, 103, 109, 127, 137, 149, 157, 167, 173, 181, 193, 199, 227, 233, 241, 257, 269, 277, 283, 307, 313, 331, 347, 353, 367, 379, 389, 401, 419, 431, 439, 449, 461, 467, 487, 499, 509, 541, 557, 569, 577
Offset: 1

Views

Author

Clark Kimberling, Jun 05 2015

Keywords

Comments

Partition of the positive integers: A258036, A258037;
Corresponding partition of the primes: A258038, A258039.

Examples

			D(prime(2), 1) = 3 - 2 > 0, so a(1) = prime(1) = 2;
D(prime(3), 2) = 5 - 2*3 + 2 > 0, so a(2) = prime(2) = 3;
D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0.

Links

Crossrefs

Cf. A258036, A258037, A258038.

Programs

  • Mathematica
    u = Table[Prime[Range[k]], {k, 1, 1000}];
v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];
w1 = Flatten[Position[v, -1]] (* A258036 *)
w2 = Flatten[Position[v, 1]]  (* A258037 *)
p1 = Prime[w1]  (* A258038 *)
p2 = Prime[w2]  (* A258039 *)

Formula

D(prime(k), k-1) = Sum_{i=0..k-1} (-1)^i*prime(k-i)*binomial(k-1,i). [corrected by Jason Yuen, Nov 13 2024]
a(n) = prime(A258037(n)). - Jason Yuen, Nov 13 2024
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.