Jeremy Rouse - cp's OEIS frontend

User: Jeremy Rouse

Jeremy Rouse has authored 3 sequences.

A259419 A Somos-4 like sequence connected with the elliptic curve y^2 + y = x^3 - 3x + 4.

1, 2, 1, -3, -7, -17, 2, 101, 247, 571, -1669, -13766, -43101, -205897, 1640929, 8217293, 101727662, 173114917, -5439590147, -70987557871, -993401657957, 2105332714614, 208894282701569, 3110590234593151, 37334338699443371, -891470356091782993, -33292234868859857114

Offset: 0

Jeremy Rouse, Jun 26 2015

All the terms of the sequence are integers. Moreover, a(n)^2 is the denominator of the x-coordinate of (2n+3)P, where P = (4,7) is the point on the elliptic curve E: y^2 + y = x^3 - 3x + 4.

			If P = (4,7), then (2*4+3)*P = (-104/49, 181/343). The denominator of the x-coordinate is 49 = a(4)^2.

Seiichi Manyama, Table of n, a(n) for n = 0..195
Alexi Block Gorman, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, and Jeremy Rouse, The density of primes dividing a particular non-linear recurrence sequence, arXiv:1508.02464 [math.NT], 2015.
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
PUMaC, PUMac 2015 Power Round. See page 21, denoted by d_n.

Cf. A006720.

a[ n_] := Module[ {v, m, s = 1}, m = If[ n < -1, s = -1; -3 - n, n] + 5; v = Join[{-2, -1, -1, 1, 1, 2, 1}, Table[0, {m - 7}]]; Do[ v[[k]] = (5 v[[k - 3]] v[[k - 4]] - v[[k - 1]] v[[k - 6]]) / v[[k - 7]], {k, 8, m}]; s v[[m]]]; (* Michael Somos, Aug 13 2015 *)

a = vector(99); a[1]=2; a[2]=1; a[3] = -3; a[4] = -7; for(n=5,#a,if(Mod(n,3)==Mod(2,3),a[n]=(a[n-1]*a[n-3]-3*a[n-2]^2)/a[n-4],a[n]=(a[n-1]*a[n-3]-a[n-2]^2)/a[n-4])); a

{a(n) = my(v, s=1); if( n<-1, n = -3-n; s = -1); n += 5; v = concat( [-2, -1, -1, 1, 1, 2, 1], vector( max(0, n-7))); for(k=8, n, v[k] = (5 * v[k-3] * v[k-4] - v[k-1] * v[k-6]) / v[k-7]); s * v[n]}; /* Michael Somos, Aug 13 2015 */

a(n) = (a(n-1)*a(n-3) - a(n-2)^2)/a(n-4) if n is not 2 mod 3, and a(n) = (a(n-1)*a(n-3) - 3*a(n-2)^2)/a(n-4) if n is 2 mod 3.
a(n) = - a(-3-n) for all n in Z. - Michael Somos, Aug 13 2015
a(n)*a(n+7) = -1*a(n+1)*a(n+6) +5*a(n+3)*a(n+4) for all n in Z. - Michael Somos, Aug 13 2015
a(n)*a(n+8) = -4*a(n+2)*a(n+6) +5*a(n+3)*a(n+5) for all n in Z. - Michael Somos, Aug 13 2015
Let t(n) be a strong elliptic divisibility sequence as given in [Kimberling, p. 16] where x = 5^(1/4), y = 3^(1/3), z = 1. Then a(n) = t(2*n + 3) / if( 3|n, y, 1). - Michael Somos, Aug 13 2015

A227199 Primes that divide some term of A006720.

Original entry on oeis.org

2, 3, 7, 11, 23, 41, 47, 53, 59, 61, 71, 73, 83, 97, 101, 113, 127, 139, 149, 157, 173, 179, 181, 191, 197, 199, 211, 223, 229, 239, 257, 263, 271, 277, 281, 307, 331, 337, 347, 359, 373, 379, 383, 389, 397, 409, 419, 433, 439, 443, 463, 467, 479, 499, 509

Offset: 1

Jeremy Rouse, Sep 18 2013

The density of primes in this sequence is 11/21.

If p divides A006720(n) for some n, then the smallest such n is <= (1/2)*((sqrt(p)+1)^2+3).

			11 is in the sequence because 11 divides A006720(10) = 1529.

Jeremy Rouse, Table of n, a(n) for n = 1..10000
R. Jones and J. Rouse, Galois Theory of Iterated Endomorphisms, Proceedings of the London Mathematical Society, 100, no. 3 (2010), 763-794.

[ n : n in [2..500] | IsPrime(n) and (n ne 37) and (Order(EllipticCurve([GF(n)!0,0,1,-1,0])![0,0,1]) mod 2 eq 1) ];

A053587 Indices of A052344 (ways to write n as sum of two nonzero triangular numbers) where record values are reached.

Original entry on oeis.org

2, 16, 81, 471, 1056, 1381, 6906, 17956, 34531, 40056, 200281, 520731, 1001406, 1482081, 7410406, 19267056, 37052031, 60765331, 303826656, 789949306, 1519133281, 3220562556, 13429138206, 16102812781, 41867313231, 80514063906, 196454315931, 711744324931

Offset: 1

Jeremy Rouse, Jan 19 2000

The subsequence of primes begins: 2, 1381, 1519133281 [Jonathan Vos Post, Feb 01 2011].

			The order of the terms is ignored when deciding in how many ways the sum can be expressed. For example, a(2) does not equal 9, although 9 = 3 + 6 = 6 + 3.
a(2) = 16 because 16 = 1 + 15 = 6 + 10. a(3) = 81 because 81 = 3 + 78 = 15 + 66 = 36 + 55.

Kenneth Korbin, Problem 665, The College Mathematics Journal Vol. 30 No. 5, Nov. 1999. Asks for the 48th number in this sequence.

Probably differs from A052348 only at n=1, 2, 4.

Cf. A000217, A052343, A052344, A052345, A052346, A052347, A052348.

More terms from Christian G. Bower, Jan 23 2000
a(25)-a(26) from Donovan Johnson, Jun 26 2010
a(27)-a(28) from Donovan Johnson, Mar 20 2013

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User: Jeremy Rouse

A259419 A Somos-4 like sequence connected with the elliptic curve y^2 + y = x^3 - 3x + 4.

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A227199 Primes that divide some term of A006720.

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A053587 Indices of A052344 (ways to write n as sum of two nonzero triangular numbers) where record values are reached.

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