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A353280
n is a term if n = 0 or n does not divide F(n, k) for all k >= 0, where F(n, k) are the Fibonacci numbers A352744.
2
0, 5, 6, 10, 12, 15, 18, 20, 24, 25, 30, 35, 36, 40, 42, 45, 48, 50, 54, 55, 56, 60, 65, 66, 70, 72, 75, 78, 80, 84, 85, 90, 91, 95, 96, 100, 102, 105, 108, 110, 112, 114, 115, 120, 125, 126, 130, 132, 135, 138, 140, 144, 145, 150, 153, 155, 156, 160, 162, 165
OFFSET
1,2
COMMENTS
n is a term if 0 is not a term of the sequence A352747(n, .). Since A352747(n, .) is for all n a pure periodic sequence, it is sufficient to require that 0 is not a term of period(A352747(n, .)). Since the length of the period is <= n, the condition can be checked in a finite number of steps.
The multiples of 5 and 6 (A093509) are a subsequence. The terms not of this form start 56, 91, 112, ..., and are in A353281.
EXAMPLE
period(A352747(6, .)) = (5, 1, 3) is zero-free, therefore 6 is a term of a.
period(A352747(7, .)) = (1, 0, 6, 5, 4, 3, 2), thus 7 is not a term of a.
MAPLE
f := n -> combinat:-fibonacci(n): F := (n, k) -> (n-1)*f(k) + f(k+1):
df := n -> denom(f(n)/n) - 1: period := n -> [seq(modp(F(k, n), n), k = 0..df(n))]:
isA353280 := n -> n = 0 or not member(0, period(n)):
select(isA353280, [$(0..166)]);
PROG
(SageMath)
def F(n, k): return (n - 1)*fibonacci(k) + fibonacci(k + 1)
def df(n): return denominator(fibonacci(n) / n)
def period(n): return (Integer(n).divides(F(k, n)) for k in range(df(n)))
def isA353280(n): return n == 0 or not any([k == True for k in period(n)])
def A353280List(upto): return [n for n in range(upto + 1) if isA353280(n)]
print(A353280List(165))
CROSSREFS
a = A093509 union A353281.
Sequence in context: A037359 A099538 A093614 * A093509 A105953 A164095
KEYWORD
nonn
AUTHOR
Peter Luschny, Apr 09 2022
STATUS
approved