A217290 -id:A217290 - cp's OEIS frontend

A060728 Numbers n such that Ramanujan's equation x^2 + 7 = 2^n has an integer solution.

3, 4, 5, 7, 15

Offset: 1

Lekraj Beedassy, Apr 25 2001

See A038198 for corresponding x. - Lekraj Beedassy, Sep 07 2004
Also numbers such that 2^(n-3)-1 is in A000217, i.e., a triangular number. - M. F. Hasler, Feb 23 2009
With respect to M. F. Hasler's comment above, all terms 2^(n-3) - 1 are known as the Ramanujan-Nagell triangular numbers (A076046). - Raphie Frank, Mar 31 2013
Interestingly enough, all the solutions correspond to noncomposite x, i.e., x = 1 for the first term, and primes 3, 5, 11, 181 for the following terms. - M. F. Hasler, Mar 11 2024

			The fifth and ultimate solution to Ramanujan's equation is obtained for the 15th power of 2, so that we have x^2 + 7 = 2^15 yielding x = 181.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 181, p. 56, Ellipses, Paris 2008.
J. Roberts, Lure of the Integers. pp. 90-91, MAA 1992.
Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts (2002): 96-98.

T. Skolem, S. Chowla and D. J. Lewis, The Diophantine Equation 2^(n+2)-7=x^2 and Related Problems. Proc. Amer. Math. Soc. 10 (1959) 663-669. [_M. F. Hasler_, Feb 23 2009]
Anonymous, Developing a general 2nd degree Diophantine Equation x^2 + p = 2^n
M. Beeler, R. W. Gosper and R. Schroeppel, HAKMEM: item 31: A Ramanujan Problem (R. Schroeppel)
Curtis Bright, Solving Ramanujan's Square Equation Computationally
Spencer De Chenne, The Ramanujan-Nagell Theorem: Understanding the Proof
T. Do, Developing A General 2nd Degree Diophantine Equation x^2 + p = 2^n
A. Engel, Problem-Solving Strategies. p. 126.
Gerry Myerson, Bibliography
T. Nagell, The Diophantine equation x^2 + 7 = 2^n, Ark. Mat. 4 (1961), no. 2-3, 185-187.
S. Ramanujan, Journal of the Indian Mathematical Society, Question 464(v,120)
Eric Weisstein's World of Mathematics, Ramanujan's Square Equation
Eric Weisstein's World of Mathematics, Diophantine Equation 2nd Powers
Wikipedia, Carmichael's Theorem
Wikipedia, Diophantine equation

Cf. A002249, A038198, A076046, A077020, A077021, A107920, A215795, A227078

[n: n in [0..100] | IsSquare(2^n-7)]; // Vincenzo Librandi, Jan 07 2014

ramaNagell[n_] := Reduce[x^2 + 7 == 2^n, x, Integers] =!= False; Select[ Range[100], ramaNagell] (* Jean-François Alcover, Sep 21 2011 *)

is(n)=issquare(2^n-7) \\ Anders Hellström, Dec 12 2015

a(n) = log_2(8*A076046(n) + 8) = log_2(A227078(n) + 7)

Empirically, a(n) = Fibonacci(c + 1) + 2 = ceiling[e^((c - 1)/2)] + 2 where {c} is the complete set of positive solutions to {n in N | 2 cos(2*Pi/n) is in Z}; c is in {1,2,3,4,6} (see A217290).

Added keyword "full", M. F. Hasler, Feb 23 2009

A242963 Numbers n such that A242962(n) = sigma(n) = A000203(n).

Original entry on oeis.org

5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Jaroslav Krizek, May 29 2014

nonn

A242962(n) = (n*(n+1)/2) mod antisigma(n) = A000217(n) mod A024816(n).
Union of number 5 and numbers >= 7.
Conjecture: this sequence lists all the positive integers n such that, for some integer k, (sin(k*Pi/n))^2 is irrational. - Lorenzo Sauras Altuzarra, Jan 27 2020

Michael De Vlieger, Table of n, a(n) for n = 1..14995

Cf. A000203, A000217, A002961, A024816, A242962, A243117, A243118, A217290, A009005.

[n: n in [3..100000] | ((n*(n+1)div 2) mod (n*(n+1)div 2-SumOfDivisors(n))) eq (SumOfDivisors(n))]

Select[Range[3, 71], DivisorSigma[1, #] == Mod[PolygonalNumber@ #, Total@ Complement[Range@ #, Divisors@ #]] &] (* Michael De Vlieger, Jan 28 2020 *)

A218129 2^(((c - 2)^2 + (c - 2))/2) + n = a(n), where c are the positive solutions to {y in N | 2cos(2Pi/y) is in Z}; c = {1,2,3,4,6}.

Original entry on oeis.org

1, 2, 4, 11, 1028

Offset: 0

Raphie Frank, Oct 21 2012

The set {c} consists of the complete set of positive solutions to the short proof of the Crystallographic Restriction Theorem {1, 2, 3, 4, 6} (see A217290).
Let {V} = prime(a(n)) = {2, 3, 7, 31, 8191}. Then all elements of {V} follow form x^2 + x + 1 for some x in R; x = {(sqrt(5) - 1)/2, 1, 2, 5, 90}. (V + V(mod 2) - 2)/2 gives the complete set of Ramanujan-Nagell triangular numbers (A076046) = {0, 1, 3, 15, 4095} == (2^F_(c + 1) - 2)/2 (see A215929); F_n the n-th Fibonacci number (A000045).
Additionally, 2*V - 1 = {3, 5, 13, 61, 16381} is prime and, therefore, all elements of {V} are links in a Cunningham chain of the 2nd kind (see A005382).

			2^(((1 - 2)^2 + (1 - 2))/2) + 0 = 2^(a(-1) - 1) + 0 = 1 = a(0).
2^(((2 - 2)^2 + (2 - 2))/2) + 1 = 2^(a(0) - 1) + 1 = 2 = a(1).
2^(((3 - 2)^2 + (3 - 2))/2) + 2 = 2^(a(1) - 1) + 2 = 4 = a(2).
2^(((4 - 2)^2 + (4 - 2))/2) + 3 = 2^(a(2) - 1) + 3 = 11 = a(3).
2^(((6 - 2)^2 + (6 - 2))/2) + 4 = 2^(a(3) - 1) + 4 = 1028 = a(4).

Eric W. Weisstein, MathWorld: Ramanujan's Square Equation
Wikipedia, Crystallographic Restriction Theorem

Cf. A217290, A215929, A076046.

to n = 4, then a(n) = 2^(a(n - 1) - 1) + n; a(-1) = 1.

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A060728 Numbers n such that Ramanujan's equation x^2 + 7 = 2^n has an integer solution.

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A242963 Numbers n such that A242962(n) = sigma(n) = A000203(n).

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A218129 2^(((c - 2)^2 + (c - 2))/2) + n = a(n), where c are the positive solutions to {y in N | 2cos(2Pi/y) is in Z}; c = {1,2,3,4,6}.

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cp's OEIS Frontend

A060728 Numbers n such that Ramanujan's equation x^2 + 7 = 2^n has an integer solution.

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Magma

Mathematica

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A242963 Numbers n such that A242962(n) = sigma(n) = A000203(n).

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Magma

Mathematica

A218129 2^(((c - 2)^2 + (c - 2))/2) + n = a(n), where c are the positive solutions to {y in N | 2*cos(2*Pi/y) is in Z}; c = {1,2,3,4,6}.

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A218129 2^(((c - 2)^2 + (c - 2))/2) + n = a(n), where c are the positive solutions to {y in N | 2cos(2Pi/y) is in Z}; c = {1,2,3,4,6}.