A025496 -id:A025496 - cp's OEIS frontend

A182535 Number of terms in Zeckendorf representation of prime(n).

1, 1, 1, 2, 2, 1, 3, 3, 2, 2, 3, 2, 3, 3, 2, 4, 3, 3, 4, 3, 3, 3, 4, 1, 2, 4, 3, 3, 4, 3, 4, 3, 4, 4, 2, 3, 2, 4, 3, 3, 3, 3, 3, 4, 5, 2, 5, 4, 5, 5, 1, 3, 2, 3, 3, 4, 3, 4, 4, 4, 4, 3, 5, 4, 5, 4, 4, 4, 5, 5, 5, 4, 5, 6, 2, 3, 4, 4, 3, 4, 3, 4, 5, 3, 4, 4, 5, 5, 4, 5, 3, 3, 3, 5, 6, 4, 5, 2, 3, 5, 4, 4, 4, 5, 5

Offset: 1

Alex Ratushnyak, May 05 2012

nonn

Alternately, the minimum number of Fibonacci numbers which sum to prime(n). - Alan Worley, Apr 17 2015

			prime(4)=7, and 7 is represented as 5+2, so a(4)=2.
prime(7)=17, and 17 is represented as 13+3+1, so a(7)=3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007895, A020908, A020910, A025496-A025502.

f[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; Count[fr, 1]]; f@# & /@ Prime@ Range@ 105 (* Robert G. Wilson v, Apr 22 2015 *)

a(n) = A007895(A000040(n)).

A182576 Number of 1's in the Zeckendorf representation of n^2.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 1, 4, 4, 4, 3, 3, 3, 4, 3, 4, 4, 3, 3, 3, 4, 5, 5, 6, 4, 5, 3, 3, 5, 3, 4, 3, 6, 4, 2, 4, 4, 5, 6, 6, 7, 3, 4, 6, 5, 4, 5, 5, 5, 5, 6, 3, 5, 7, 4, 5, 6, 4, 6, 4, 5, 6, 5, 6, 6, 6, 4, 6, 7, 7, 8, 5, 6, 7, 6, 6, 7, 4, 4, 6, 3

Offset: 0

Alex Ratushnyak, May 05 2012

nonn

Also the minimum number of different Fibonacci numbers that sum up to n^2. - Carmine Suriano, Jul 03 2013

See A088060 for some comments related to the occurrence of 2's. - Peter Munn, Mar 22 2021

			a(11)=4 since 11^2 = 121 = 89 + 21 + 8 + 3 = fib(11) + fib(8) + fib(6) + fib(4). - _Carmine Suriano_, Jul 03 2013

Carmine Suriano, Table of n, a(n) for n = 0..1000

Cf. A007895, A020908, A020910, A025496-A025502, A088060, A182535, A182580.

A182577 Number of ones in Zeckendorf representation of n!

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 3, 5, 6, 9, 8, 11, 11, 11, 16, 17, 17, 18, 23, 23, 28, 31, 33, 27, 33, 29, 40, 37, 42, 42, 41, 44, 47, 44, 53, 56, 57, 50, 64, 55, 59, 68, 63, 72, 70, 61, 69, 85, 80, 83, 87, 97, 98, 101, 87, 91, 100, 102, 114, 108, 116, 109, 117, 117, 113, 124

Offset: 0

Alex Ratushnyak, May 05 2012

nonn

			5! = {1, 0, 0, 1, 0, 1, 0, 0, 1, 0} in the Zeckendorf base.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A007895, A020908, A020910, A025496-A025502, A182535, A182576, A000142.

from math import factorial
def A182577(n):
    m, tlist, s = factorial(n), [1,2], 0
    while tlist[-1]+tlist[-2] <= m:
        tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        if d <= m:
            s += 1
            m -= d
    return s # Chai Wah Wu, Jun 15 2018

A182578 Number of ones in Zeckendorf representation of n^n.

Original entry on oeis.org

1, 1, 2, 3, 3, 6, 3, 10, 13, 12, 16, 15, 20, 24, 20, 30, 25, 31, 26, 33, 33, 31, 34, 42, 49, 49, 53, 55, 56, 55, 58, 64, 64, 67, 73, 78, 70, 76, 77, 75, 89, 83, 92, 90, 106, 99, 100, 99, 107, 116, 107, 115, 125, 125, 122, 119, 127, 137, 127, 138, 155, 156, 153, 160

Offset: 0

Alex Ratushnyak, May 05 2012

nonn

			5^5 = {1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0} in the Zeckendorf base.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A007895, A020908, A020910, A025496-A025502, A182535, A182576, A182577, A000312.

def A182578(n):
    m, tlist, s = n**n, [1,2], 0
    while tlist[-1]+tlist[-2] <= m:
        tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        if d <= m:
            s += 1
            m -= d
    return s # Chai Wah Wu, Jun 14 2018

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A182535 Number of terms in Zeckendorf representation of prime(n).

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A182576 Number of 1's in the Zeckendorf representation of n^2.

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A182577 Number of ones in Zeckendorf representation of n!

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A182578 Number of ones in Zeckendorf representation of n^n.

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