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.

A073272 A000040(n+1) - A073271(n).

Original entry on oeis.org

0, 1, 0, 3, -1, 3, -1, 0, 5, -3, 3, 3, -1, -1, 1, 5, -3, 3, 3, -3, 3, -1, -1, 5, 3, -1, 3, -1, -9, 11, -1, 5, -7, 9, -3, 1, 3, -1, 1, 5, -7, 9, -1, 3, -9, 1, 9, 3, -1, -1, 5, -7, 5, 1, 1, 5, -3, 3, 3, -7, -3, 11, 3, -1, -9, 9, -3, 9, -1, -1, -1, 3, 1, 3, -1, -1, 5, -3, -1, 9, -7, 9, -3, 3, -1
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2002

Keywords

Comments

Observation/conjecture: a(n)=0 iff A073271(n) in {3, 7, 23}.

Examples

			For n=11, A000040(11)*A000040(13)/A000040(12) = 31*41/37 = 1271/37 = (34*37+13)/37, therefore A073271(11)=34; a(11) = A000040(12)-A073271(11) = 37-34 = +3.

Links

Crossrefs

Cf. A073274.

Programs

  • Magma
    [NthPrime(n+1)-Floor(NthPrime(n)*NthPrime(n+2) / NthPrime(n+1)): n in [1..80]]; // Vincenzo Librandi, May 31 2015
  • Mathematica
    Table[Prime[n+1] - Floor[Prime[n] Prime[n+2] / Prime[n+1]], {n, 80}] (* Vincenzo Librandi, May 31 2015 *)
  • PARI
    a(n,p=prime(n))=my(q=nextprime(p+1),r=nextprime(q+1)); q - p*r\q \\ Charles R Greathouse IV, Jun 02 2015

A073273 a(n) = floor(sqrt(prime(n)*prime(n+2))).

Original entry on oeis.org

3, 4, 7, 9, 13, 15, 19, 23, 26, 32, 35, 39, 43, 47, 52, 56, 62, 65, 69, 74, 77, 83, 89, 94, 99, 103, 105, 109, 117, 121, 131, 134, 142, 144, 152, 156, 161, 167, 172, 176, 184, 186, 193, 195, 203, 210, 218, 225, 229, 233, 236, 244, 248, 256, 262, 266, 272, 275
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2002

Keywords

Comments

A000040(n) < a(n) < A000040(n+2).

Examples

			prime(10)*prime(12) = 29*37 = 1073 = 32*32+49, therefore a(10)=32; A073274(10) = prime(11)-a(10) = 31-32 = -1.

Links

Crossrefs

Cf. A000040, A073271, A073274, A078444.

Programs

  • Magma
    [Floor(Sqrt(NthPrime(n)*NthPrime(n+2))): n in [1..60]]; // Vincenzo Librandi, Dec 12 2015
  • Mathematica
    Table[Floor[Sqrt[Prime[n] Prime[n + 2]]], {n, 60}] (* Vincenzo Librandi, Dec 12 2015 *)
  • PARI
    a(n) = sqrtint(prime(n)*prime(n+2)); \\ Michel Marcus, Dec 12 2015

Formula

a(n) = A098090(A028310(n - 1)) + A089038(n). - Miko Labalan, Dec 12 2015

A022462 a(n) = prime(n)*prime(n+2) mod prime(n+1).

Original entry on oeis.org

1, 1, 6, 3, 5, 9, 11, 22, 17, 19, 13, 33, 35, 23, 17, 47, 49, 43, 63, 61, 55, 59, 41, 65, 93, 95, 99, 101, 57, 71, 107, 125, 119, 129, 139, 121, 139, 143, 137, 167, 161, 171, 185, 189, 175, 67, 175, 219, 221, 209, 227, 221, 191, 221, 227, 257, 259, 253, 273, 263
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A090076, A073271, A000040.

Programs

  • Magma
    [NthPrime(n)*NthPrime(n+2) mod NthPrime(n+1): n in [1..50]]; // G. C. Greubel, Feb 28 2018
  • Mathematica
    Table[Mod[Prime[n]Prime[n+2], Prime[n+1]], {n, 1, 50}] (* G. C. Greubel, Feb 28 2018 *)
  • PARI
    a(n) = (prime(n)*prime(n+2)) % prime(n+1); \\ Michel Marcus, Sep 30 2013

Formula

a(n) = A090076(n) modulo A000040(n+1). - Michel Marcus, Sep 30 2013
Conjecture: a(n) = prime(n)*prime(n+2) - prime(n+1)*(prime(n) + prime(n+2) - prime(n+1) - 1), except for a(3) and a(8). - Ridouane Oudra, Oct 26 2021

Extensions

Edited by Reinhard Zumkeller, Jul 22 2002

A258326 a(1) = 3; for n > 1, a(n) = a(n-1) + prime(n+2) - 2*prime(n+1) + 2*prime(n) - prime(n-1).

Original entry on oeis.org

3, 4, 8, 8, 14, 14, 20, 24, 24, 34, 34, 38, 44, 48, 52, 54, 64, 64, 68, 76, 76, 84, 90, 92, 98, 104, 104, 110, 122, 116, 132, 132, 146, 140, 154, 156, 160, 168, 172, 174, 188, 182, 194, 194, 208, 210, 214, 224, 230, 234, 234, 248, 246, 256, 262, 264, 274, 274
Offset: 1

Views

Author

Gionata Neri, May 26 2015

Keywords

Comments

Conjecture: except for a(3)=8 and a(8)=24, this is the same as A073271.

Programs

  • Mathematica
    f[n_] := Block[{a = {3}}, g[x_] := a[[x - 1]] + Prime[x + 2] - 2 Prime[x + 1] + 2 Prime@ x - Prime[x - 1]; Do[AppendTo[a, g@ k], {k, 2, n}]; a]; f@ 60 (* Michael De Vlieger, Jun 02 2015 *)
RecurrenceTable[{a[1]==3,a[n]==a[n-1]+Prime[n+2]-2Prime[n+1]+2Prime[n]-Prime[n-1]},a,{n,60}] (* Harvey P. Dale, Mar 25 2019 *)
  • PARI
    v=[3];n=2;while(n<50,v=concat(v,v[#v]+prime(n+2) - 2*prime(n+1)+2*prime(n)-prime(n-1));n++);v \\ Derek Orr, May 30 2015

Formula

For n>1, a(n) = a(n-1) - A062234(n+1) + A062234(n). - Michel Marcus, May 31 2015
Showing 1-4 of 4 results.

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