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.

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

Author

Peter Luschny, Apr 09 2022

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

a = A093509 union A353281.

Cf. A352744, A352747.

Programs

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)]);

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))