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 1-2 of 2 results.

A167053 a(1)=3; for n > 1, a(n) = 1 + a(n-1) + gcd( a(n-1)*(a(n-1)+2), A073829(a(n-1)) ).

Original entry on oeis.org

3, 19, 39, 81, 165, 333, 335, 673, 1347, 1349, 1351, 1353, 1355, 1357, 1359, 2721, 2723, 2725, 2727, 5457, 5459, 5461, 5463, 5465, 5467, 5469, 10941, 10943, 10945, 10947, 21897, 21899, 21901, 21903, 21905, 21907, 21909, 43821, 43823, 43825, 43827, 43829, 43831
Offset: 1

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Author

Vladimir Shevelev, Oct 27 2009

Keywords

Comments

The first differences are 16, 20, 42, etc. They are either 2 or in A075369 or in A008864, see A167054.
A proof follows from Clement's criterion of twin primes.

Examples

			a(2) = 1 + 3 + gcd(3*5, 4*(2! + 1) + 3) = 19.

References

  • E. Trost, Primzahlen, Birkhäuser-Verlag, 1953, pages 30-31.

Links

Crossrefs

Cf. A073829, A008864, A167054.
Cf. A166944, A166945, A116533, A163961, A163963, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.

Programs

  • Maple
    A073829 := proc(n) n+4*((n-1)!+1) ; end proc:
A167053 := proc(n) option remember ; local aprev; if n = 1 then 3; else aprev := procname(n-1) ; 1+aprev+gcd(aprev*(aprev+2),A073829(aprev)) ; end if; end proc:
seq(A167053(n),n=1..60) ; # R. J. Mathar, Dec 17 2009
  • Mathematica
    A073829[n_] := 4((n-1)! + 1) + n;
a[1] = 3;
a[n_] := a[n] = 1 + a[n-1] + GCD[a[n-1] (a[n-1] + 2), A073829[a[n-1]]];
Array[a, 60] (* Jean-François Alcover, Mar 25 2020 *)

Extensions

Definition shortened and values from a(4) on replaced by R. J. Mathar, Dec 17 2009

A073830 a(n) = 4*((n-1)! + 1) + n (mod n*(n + 2)).

Original entry on oeis.org

0, 2, 0, 8, 0, 10, 56, 12, 22, 14, 0, 16, 182, 18, 34, 20, 0, 22, 380, 24, 46, 26, 552, 28, 29, 30, 58, 32, 0, 34, 992, 36, 37, 38, 74, 40, 1406, 42, 82, 44, 0, 46, 1892, 48, 94, 50, 2256, 52, 53, 54, 106, 56, 2862, 58, 59, 60, 118, 62, 0, 64, 3782, 66, 67, 68, 134, 70
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 12 2002

Keywords

Links

Crossrefs

Cf. A001359, A005563, A006512, A073831, A073832.

Programs

  • Magma
    [(4*(Factorial(n-1)+1)+n) mod (n^2+2*n): n in [1..70]]; // Vincenzo Librandi, May 04 2014
  • Maple
    A073830 := proc(n)
    4*((n-1)!+1)+n ;
    modp(%,n*(n+2)) ;
end proc:
seq(A073830(n),n=1..60) ; # R. J. Mathar, Feb 21 2017
  • Mathematica
    a[n_] := Mod[4 ((n - 1)! + 1) + n, n (n + 2)]; Array[a, 66] (* Jean-François Alcover, Feb 22 2018 *)

Formula

a(n) = A073829(n) mod A005563(n).
For n > 1: a(n) = 0 iff (n, n+2) are twin primes (Clement, 1949).
From Bernard Schott, Nov 16 2021: (Start)
If p is an odd prime, and p+2 is composite, then a(p) = p*(p+1).
If m is composite, and m+2 is prime, then a(m) = 2*(m+2).
If n even >= 4, a(n) = n + 4.
If p prime >= 5, a(p^2) = p^2 + 4. (End)
Showing 1-2 of 2 results.

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