A353987 -id:A353987 - cp's OEIS frontend

A353986 Numbers k such that Fibonacci(k) and Fibonacci(k+1) have the same binary weight (A000120).

1, 2, 4, 7, 24, 27, 49, 51, 52, 69, 75, 114, 130, 131, 158, 169, 186, 217, 250, 263, 292, 335, 340, 345, 374, 474, 500, 507, 520, 547, 565, 583, 600, 604, 627, 717, 760, 791, 828, 831, 908, 996, 997, 1011, 1023, 1061, 1081, 1114, 1242, 1641, 1660, 1763, 1780

Offset: 1

Amiram Eldar, May 13 2022

Numbers k such that A011373(k) = A011373(k+1).

The corresponding values of A011373(k) are 1, 1, 2, 3, 6, 11, 18, 17, 17, 23, 23, 43, 42, 42, 51, ...

			1 is a term since A011373(1) = A011373(2) = 1.
4 is a term since A011373(4) = A011373(5) = 2.

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

Cf. A000045, A000120, A011373.

A353987 is a subsequence.

s[n_] := s[n] = DigitCount[Fibonacci[n], 2, 1]; Select[Range[2000], s[#] == s[# + 1] &]

isok(k) = hammingweight(fibonacci(k)) == hammingweight(fibonacci(k+1)); \\ Michel Marcus, May 13 2022

from itertools import islice
def A353986_gen(): # generator of terms
        a, b, k, ah = 1, 1, 1, 1
        while True:
            if ah == (bh := b.bit_count()):
                yield k
            a, b, ah = b, a+b, bh
            k += 1
A353986_list = list(islice(A353986_gen(),30)) # Chai Wah Wu, May 13 2022

A354301 Numbers k such that k!, (k+1)! and (k+2)! have the same binary weight (A000120).

Original entry on oeis.org

0, 7, 12, 262, 12887667

Offset: 1

Amiram Eldar, May 23 2022

Numbers k such that A079584(k) = A079584(k+1) = A079584(k+2).

The corresponding values of A079584(k) are 1, 6, 12, 747, 136453086, ...

			7 is a term since A079584(7) = A079584(8) = A079584(9) = 6.
12 is a term since A079584(12) = A079584(13) = A079584(14) = 12.

Subsequence of A354300.

Cf. A000120, A000142, A079584, A353987.

s[n_] := s[n] = DigitCount[n!, 2, 1]; Select[Range[0, 300], s[#] == s[# + 1] == s[# + 2] &]

from itertools import count, islice
def wt(n): return bin(n).count("1")
def agen(): # generator of terms
    n, fn, fn1, fn2, wtn, wtn1, wtn2 = 0, 1, 1, 2, 1, 1, 1
    for n in count(0):
        if wtn == wtn1 == wtn2: yield n
        fn, fn1, fn2 = fn1, fn2, fn2*(n+3)
        wtn, wtn1, wtn2 = wtn1, wtn2, wt(fn2)
print(list(islice(agen(), 4))) # Michael S. Branicky, May 23 2022

A374961 Numbers k such that 2^k, 2^(k+1) and 2^(k+2) have the same number of terms in their Zeckendorf representation (A007895).

Original entry on oeis.org

5, 6, 1931, 4127, 26584

Offset: 1

Amiram Eldar, Jul 25 2024

Numbers k such that A020908(k) = A020908(k+1) = A020908(k+2).
The corresponding values of A020908(k) are 3, 3, 763, 1660, 10596, ... .
a(6) > 10^5, if it exists.

			5 is a term since A020908(5) = A020908(6) = A020908(7) = 3.
763 is a term since A020908(1931) = A020908(1932) = A020908(1933) = 763.

Cf. A007895, A020908, A353987.

Subsequence of A374960.

z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
s[n_] := s[n] = z[2^n]; Select[Range[0, 4200], s[#] == s[# + 1] == s[# + 2] &]

A007895(n)=if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s); \\ Charles R Greathouse IV at A007895
lista(kmax) = {my(z1 = A007895(1), z2 = A007895(2), z3); for(k = 2, kmax, z3 = A007895(2^k); if(z1 == z2 && z2 == z3, print1(k-2 , ", ")); z1 = z2; z2 = z3);}

cp's OEIS Frontend

A353986 Numbers k such that Fibonacci(k) and Fibonacci(k+1) have the same binary weight (A000120).

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A354301 Numbers k such that k!, (k+1)! and (k+2)! have the same binary weight (A000120).

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A374961 Numbers k such that 2^k, 2^(k+1) and 2^(k+2) have the same number of terms in their Zeckendorf representation (A007895).

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