A247024 -id:A247024 - cp's OEIS frontend

A076046 Ramanujan-Nagell numbers: the triangular numbers (of the form a*(a+1)/2) which are also of the form 2^b - 1.

0, 1, 3, 15, 4095

Offset: 1

Burt Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002

Ramanujan conjectured and Nagell proved, that the given numbers are the only ones. This sequence is equivalent to A060728, the list of numbers n such that x^2 + 7 = 2^n is soluble, by changing from n to 2^(n-3)-1.
These 5 numbers are therefore the only ones which appear in column k=2 and also in the first subdiagonal of the Stirling2 Sheffer matrix S(n,k) = A048993(n,k). These entries are 0 = S(0, 2) = S(1, 2) = S(1, 0), 1 = S(2, 2) = S(2, 1), 3 = S(3, 2) (intersection of the column k=2 with the first subdiagonal), 15 = S(5, 2) = S(6, 5) and 4095 = S(13, 2) = S(91, 90). The motivation to look into this came from a comment of R. J. Cano on A247024. - Wolfdieter Lang, Oct 16 2014
Named after the Indian mathematician Srinivasa Ramanujan (1887-1920) and the Norwegian mathematician Trygve Nagell (1895-1988). - Amiram Eldar, Jun 22 2021

			4095 can be written as 90*(90+1)/2, but also as 2^12 - 1.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd, 1999. See Chapter 6.
T. Nagell. The Diophantine equation x^2 + 7 = 2^n. Nordisk Mat. Tidskr., Vol. 30 (1948), pp. 62-64; Ark. Math., Vol. 4 (1960), pp. 185-187.

Yann Bugeaud and T. N. Shorey. On the number of solutions of the generalized Ramanujan-Nagell equation, J. reine angew. Math., Vol. 539 (2001), pp. 55-74.
Helmut Hasse, Uber eine diophantische Gleichung von Ramanujan-Nagell und ihre Verallgemeinerung, Nagoya Math. J., Vol. 27 (1966), pp. 77-102.
Eric Weisstein's World of Mathematics, Ramanujan's Square Equation.

Cf. A060728, A038198, A180445, A215797.

Reap[For[b = 0, b <= 12, b++, If[IntegerQ[(Sqrt[2^(b + 3) - 7] - 1)/2], Sow[2^b - 1]]]][[2, 1]] (* Jean-François Alcover, Jul 05 2017 *)
Select[Accumulate[Range[0,200]],IntegerQ[Log[2,#+1]]&] (* Harvey P. Dale, Aug 27 2019 *)

A245014 Least prime p such that 2n*4^n divides p + 4n^2 + 1.

Original entry on oeis.org

3, 47, 347, 6079, 10139, 147311, 687931, 18874111, 37748411, 104857199, 276823579, 805305791, 29662117211, 30064770287, 64424508539, 2473901161471, 11098195491707, 7421703486191, 83562883709531, 527765581330879, 369435906930971, 27866022694353007, 19421773393033147

Offset: 1

R. J. Cano Sep 17 2014

nonn

All those terms such that 2n*4^n is equal to p + 4n^2 + 1 belong to A247024.

Charles R Greathouse IV, Table of n, a(n) for n = 1..500

Cf. A247024.

a[n_] := With[{k = n*2^(2*n+1)}, p = -4*n^2-1; While[!PrimeQ[p += k]]; p]; Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Oct 09 2014, translated from Charles R Greathouse IV's PARI code *)

search(u)={ /* Slow, u must be a small integer. */
  my(log2=log(2),q,t,t0,L1=List());
  forprime(y=3,prime(10^u),
    t=log(y+1)\log2;
    while(t>t0,
      q=4*t^2+y+1;
      if(q%(t*(2^(2*t+1)))==0,
        listput(L1,[t,y]);
        t0=t;
        break
      ,
        t--
      )));
  L1
}

a(n)=my(k=n<<(2*n+1),p=-4*n^2-1); while(!isprime(p+=k),); p \\ Charles R Greathouse IV, Sep 18 2014

a(n) << n^5*1024^n by Xylouris' version of Linnik's theorem. - Charles R Greathouse IV, Sep 18 2014

a(10)-a(23) from Charles R Greathouse IV, Sep 18 2014

cp's OEIS Frontend

A076046 Ramanujan-Nagell numbers: the triangular numbers (of the form a*(a+1)/2) which are also of the form 2^b - 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica