A215797 -id:A215797 - 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 *)

A215795 Numbers n such that 2^n-1 is a triangular number (A000217).

Original entry on oeis.org

0, 1, 2, 4, 12

Offset: 1

V. Raman, Aug 23 2012

Aside from a(2), all terms are even. Probably complete; no more terms up to 10^6. - Charles R Greathouse IV, Sep 07 2012
This sequence maps to the Ramanujan-Nagell squares (8*(2^n - 1) + 1) and is therefore complete. - Raphie Frank, Sep 10 2012
Define equivalence classes on a specified real interval with respect to the symmetric transitive closure of R(x,y) = "x is an integer multiple of y". If any equivalence class is finite (the conditions for which are given in A328129), then a smallest equivalence class has cardinality 1, 2, 4 or 12. - Peter Munn, Jun 02 2021

Eric Weisstein's World of Mathematics, Ramanujan's Square Equation

Cf. A000217, A060728, A038198.
Cf. A076046 (triangular numbers of the form 2^n - 1).
Cf. A060728 (a(n) + 3).
Cf. A038198 (sqrt(8*(2^n - 1)+1)).
Cf. A215797 ((sqrt(8*(2^n - 1)+1) - 1)/2).
Cf. A328129.

Select[Range[0,15],OddQ[Sqrt[8(2^#-1)+1]]&] (* Harvey P. Dale, Dec 13 2024 *)

is(n)=issquare(8<Charles R Greathouse IV, Sep 07 2012

Four cross-references to the Ramanujan-Nagell problem added by Raphie Frank, Sep 10 2012

A374796 Positive integers that can be written as a Stirling number of the second kind in more than one way.

Original entry on oeis.org

1, 15, 4095, 66066

Offset: 1

Pontus von Brömssen, Jul 20 2024

k*(k+1)/2 is a term if k is a term of A215797 and k != 0,2.

a(5) > 10^8 (if it exists).

			With S2(n,k) = A008277(n,k):
      1 = S2( 1, 1) = S2(  2,  1) (for example);
     15 = S2( 5, 2) = S2(  6,  5);
   4095 = S2(13, 2) = S2( 91, 90);
  66066 = S2(14,11) = S2(364,363).

Cf. A008277, A024412, A215797.

A211202 Positive numbers n such that Lambda_n = A002336(n) is divisible by n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 12, 15, 16, 18, 20, 21, 22, 23, 24

Offset: 1

Raphie Frank, Feb 18 2013

nonn

Observations:
For all n in this sequence to n = 24, then y = Lambda_n/n follows form: y = (x^2 + x^k) - (floor[z^2/4]) or y = (x^2 + x^k) + (floor[z^2/4]); k = 1 or 2 and z = 0, 1, 3, 6 or 7. y (= A222786) gives the average number of spheres/dimension of the laminated lattice Kissing numbers in A222785.
e.g. Where T_x is the x-th triangular number = (1/2*(x^2 + x)), 2*T_x is the x-th pronic number = (x^2 + x) = floor[(2*x + 1)^2/4], and S_x is the x-th square = (x^2) = floor[(2*x)^2/4]:
For k = 1, z = 0 or 1, then n = {1, 4, 6, 8, 15, 20, 24}, x = {1, 2, 3, 5, 12, 29, 90}, and y = 2*T_x = {2, 6, 12, 30, 156, 870, 8190}.
For k = 2, z = 0 or 1, then n = {1, 5, 7, 23}, x = {1, 2, 3, 45}, and y = 2*T_x + 2*T_(-x) = 2*S_x = {2, 8, 18, 4050}.
For k = 1, z = 3, then n = {3, 7, 12, 16}, x = {2, 4, 7, 16}, and y = 2*T_x - 2*T_1 = {4, 18, 54, 270}.
For k = 1, z = 6, then n = {2, 18}, x = {3, 20}, and y = 2*T_x - S_3 = {3, 411}.
For k = 1, z = 7, then n = {5, 7, 8, 21}, x = {4, 5, 6, 36}, and y = 2*T_x - 2*T_3 = {8, 18, 30, 1320}.
For k = 1, z = 7, then n = {6, 7, 12, 22}, x = {0, 2, 6, 47}, and y = 2*T_x + 2*T_3 = {12, 18, 54, 2268}.
For the special case where k = 1 and z = 0 or 1, then all associated x values follow form (A216162(n) - A216162(n - 2)) [type 1] or (A216162(n) - A216162(n - 1)) [type II] for some n in N. Type II x values = {1, 2, 5, 90} (= A215797(n+1)) are associated with the positive Ramanujan-Nagell triangular numbers = {1, 3, 15, 4095} (= A076046(n+1)) by the formula 1/2*(x^2 + x) = T_x.

			Lambda_6/6 = 72/6 = 12, so 6 is in this sequence.
Lambda_12/12 = 648/12 = 54, so 12 is in this sequence.
Lambda_18/18 = 7398/18 = 411, so 18 is in this sequence.
Lambda_24/24 = 196560/24 = 8190, so 24 is in this sequence.
But...
Lambda_19/19 = 10668/19 = 561.47368..., so 19 is not in this sequence.

Cf. A222785, A222786, A002336, A216162

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.

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A215795 Numbers n such that 2^n-1 is a triangular number (A000217).

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A374796 Positive integers that can be written as a Stirling number of the second kind in more than one way.

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A211202 Positive numbers n such that Lambda_n = A002336(n) is divisible by n.

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