A263537 Integers k such that A098531(k) is divisible by A000071(k+2).

1, 2, 13, 25, 31, 43, 55, 61, 73, 85, 91, 103, 115, 121, 133, 145, 151, 163, 175, 181, 193, 205, 211, 223, 235, 241, 253, 265, 271, 283, 295, 301, 313, 325, 331, 343, 355, 361, 373, 385, 391, 403, 415, 421, 433, 445, 451, 463, 475, 481, 493, 505, 511, 523, 535, 541, 553

Offset: 1

Author

Altug Alkan, Oct 20 2015

Keywords

• nonn
• easy

Comments

Sequence is interesting because of the values of a(n) - a(n-1) that are 12 or 6 for n > 3.

a(2) = 2 is the only even term.

Examples

			a(1) = 1 because 1^5 mod 1 = 0.
a(2) = 2 because (1^5 + 1^5) mod (1 + 1) = 0.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A000045, A000071, A098531.

Programs

Magma

I:=[1,2,13,25,31,43]; [n le 6 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Nov 20 2015

Mathematica

lim = 560; s = Accumulate[Fibonacci[Range@lim]^5]; t = Fibonacci@ Range[2 lim] - 1; Select[Range@ lim, Divisible[s[[#]], t[[# + 2]]] &] (* Michael De Vlieger, Nov 19 2015, after Harvey P. Dale at A098531 and A000071 *)

PARI

for(n=1, 1e3, if(sum(k=1, n, fibonacci(k)^5) % sum(k=1, n, fibonacci(k)) == 0, print1(n", ")));

PARI

Vec(x*(x^5+5*x^4+11*x^3+11*x^2+x+1)/((x-1)^2*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 20 2015

Formula

From Colin Barker, Oct 20 2015: (Start)

a(n) = a(n-1)+a(n-3)-a(n-4) for n>6.

G.f.: x*(x^5+5*x^4+11*x^3+11*x^2+x+1) / ((x-1)^2*(x^2+x+1)).

(End)