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 A339669 Number of Fibonacci divisors of Lucas(n)^2 + 1. 2
 2, 2, 3, 1, 3, 2, 3, 2, 5, 1, 5, 2, 4, 2, 5, 1, 5, 2, 4, 2, 6, 1, 6, 2, 4, 2, 6, 1, 6, 2, 4, 2, 6, 1, 7, 2, 5, 2, 6, 1, 6, 2, 4, 2, 7, 1, 7, 2, 5, 2, 7, 1, 6, 2, 5, 2, 7, 1, 6, 2, 4, 2, 8, 1, 9, 2, 5, 2, 6, 1, 6, 2, 4, 2, 7, 1, 9, 2, 6, 2, 7, 1, 7, 2, 5, 2, 7, 1, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Particular attention must be paid to the regularity properties of the number of divisors of Lucas(n)^2 + 1 observed for n < 156, when a(n) = 1 or 2. From this observation, we propose two conjectures verified for n < 156. Conjecture 1: a(6*n+3) = 1. Conjecture 2: a(6*n+1) = a(6*n+5) = 2. The table in the links shows an array where terms are arranged in a table of 12 columns and 13 rows. We see the periods when a(n) = 1 and 2. LINKS Michel Lagneau, Table EXAMPLE a(8) = 5 because the divisors of Lucas(8)^2 + 1 = 47^2 + 1 = 2210 are {1, 2, 5, 10, 13, 17, 26, 34, 65, 85, 130, 170, 221, 442, 1105, 2210} with 5 Fibonacci divisors: 1, 2, 5, 13 and 34. MAPLE with(combinat, fibonacci):nn:=100:F:={}: Lucas:=n->2*fibonacci(n-1)+fibonacci(n): for k from 0 to nn do:   F:=F union {fibonacci(k)}: od:    for m from 0 to 90 do:     l:=Lucas(m)^2+1:d:=numtheory[divisors](l):n0:=nops(d):     lst:= F intersect d: n1:=nops(lst):printf(`%d, `, n1):    od: MATHEMATICA Array[DivisorSum[LucasL[#]^2 + 1, 1 &, AnyTrue[Sqrt[5 #^2 + 4 {-1, 1}], IntegerQ] &] &, 89, 0] (* Michael De Vlieger, Dec 12 2020 *) PROG (PARI) a(n) = { my(l2 = 5*fibonacci(n)^2 + 4*(-1)^n + 1, k = 1, m = 2, res = 1, g); while(m <= l2, if(l2 % m == 0, res++); g = m; m += k; k = g; ); res } \\ David A. Corneth, Dec 12 2020 CROSSREFS Cf. A000032, A000045, A001254, A339461. Sequence in context: A233431 A160650 A304092 * A171691 A088904 A241568 Adjacent sequences:  A339666 A339667 A339668 * A339670 A339671 A339672 KEYWORD nonn AUTHOR Michel Lagneau, Dec 12 2020 STATUS approved

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Last modified May 18 10:19 EDT 2021. Contains 343995 sequences. (Running on oeis4.)