A110766 -id:A110766 - cp's OEIS frontend

A110779 Fractalization of e.

2, 2, 7, 2, 1, 7, 8, 2, 2, 1, 8, 7, 1, 8, 8, 2, 2, 2, 8, 1, 4, 8, 5, 7, 9, 1, 0, 8, 4, 8, 5, 2, 2, 2, 3, 2, 5, 8, 3, 1, 6, 4, 0, 8, 2, 5, 8, 7, 7, 9, 4, 1, 7, 0, 1, 8, 3, 4, 5, 8, 2, 5, 6, 2, 6, 2, 2, 2, 4, 3, 9, 2, 7, 5, 7, 8, 5, 3, 7, 1, 2, 6, 4, 4, 7, 0, 0, 8, 9, 2, 3, 5, 6, 8, 9, 7, 9, 7, 9, 9, 5, 4, 9, 1, 5

Offset: 1

Alexandre Wajnberg, Sep 15 2005

Self-descriptive sequence: even terms are the sequence itself, odd terms are the digits of the decimal expansion of e.

Clark Kimberling, Fractal sequences.

Cf. A001113, A003602.

Cf. A110766 (of Pi), A110812 (of sqrt 2), A382130 (of phi).

a(2n) = a(n); a(2n+1) = digits of e.

A110812 Fractalization of sqrt 2.

Original entry on oeis.org

1, 1, 4, 1, 1, 4, 4, 1, 2, 1, 1, 4, 3, 4, 5, 1, 6, 2, 2, 1, 3, 1, 7, 4, 3, 3, 0, 4, 9, 5, 5, 1, 0, 6, 4, 2, 8, 2, 8, 1, 0, 3, 1, 1, 6, 7, 8, 4, 8, 3, 7, 3, 2, 0, 4, 4, 2, 9, 0, 5, 9, 5

Offset: 1

Alexandre Wajnberg, Sep 15 2005

Self-descriptive sequence: even terms are the sequence itself, odd terms are the digits of the decimal expansion of sqrt 2.

Clark Kimberling, Fractal sequences.

Cf. A002193 (sqrt 2), A003602.

Cf. A110766 (of Pi), A110779 (of e), A382130 (of phi).

a(2n)=a(n); a(2n+1)=digits of sqrt 2.

A110963 Fractalization of Kimberling's paraphrases sequence beginning with 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 4, 1, 1, 1, 5, 3, 3, 2, 6, 2, 2, 1, 7, 4, 4, 1, 8, 1, 1, 1, 9, 5, 5, 3, 10, 3, 3, 2, 11, 6, 6, 2, 12, 2, 2, 1, 13, 7, 7, 4, 14, 4, 4, 1, 15, 8, 8, 1, 16, 1, 1, 1, 17, 9, 9, 5, 18, 5, 5, 3, 19, 10, 10, 3, 20, 3, 3, 2, 21, 11, 11, 6, 22, 6, 6, 2, 23, 12, 12, 2, 24, 2, 2, 1, 25, 13

Offset: 1

Alexandre Wajnberg, Sep 26 2005

Self-descriptive sequence: terms at even indices are the sequence itself, terms at odd indices (the skeleton of this sequence) are the terms of Kimberling's paraphrases sequence (A003602) beginning with 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Clark Kimberling, Fractal sequences.
Index entries for sequences related to binary expansion of n

One more than A110962 (but note the different starting offsets).
Cf. A000265, A003602, A005408, A089309, A110812, A110779, A110766, A351565 [= 2*a(n) - 1].
Cf. A353366 (Dirichlet inverse), A353367 (sum with it).

A003602(n) = (1+(n>>valuation(n,2)))/2;
A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2)); \\ Antti Karttunen, Apr 03 2022

a(n) = n>>=valuation(n,2); 1+n>>valuation(2*n+2,2); \\ Ruud H.G. van Tol, Jun 23 2024

def A110963(n): return (1+(m:=n>>(~n&n-1).bit_length())>>(m+1&-m-1).bit_length())+1 # Chai Wah Wu, Jan 04 2024

For even n, a(n) = a(n/2), for odd n, a(n) = A003602((1+n)/2). - Antti Karttunen, Apr 03 2022
For n >= 0, (Start)
a(4n+2) = a(4n+3) = A003602(1+n).
a(8n+1) = A005408(n) = 2*n + 1.
a(4n+1) = a(8n+2) = a(8n+3) = 1+n.
a(n) = A110962(n-1) + 1.
(End)
a(n) = A353367(4*n). - Antti Karttunen, Apr 20 2022
a(n) = A003602(A003602(n)). - Ridouane Oudra, Dec 28 2024

Entry edited, starting offset corrected (from 0 to 1), and the offsets in formulas changed accordingly, and more terms added by Antti Karttunen, Apr 03 2022

A382130 Fractalization of the golden ratio.

Original entry on oeis.org

1, 1, 6, 1, 1, 6, 8, 1, 0, 1, 3, 6, 3, 8, 9, 1, 8, 0, 8, 1, 7, 3, 4, 6, 9, 3, 8, 8, 9, 9, 4, 1, 8, 8, 4, 0, 8, 8, 2, 1, 0, 7, 4, 3, 5, 4, 8, 6, 6, 9, 8, 3, 3, 8, 4, 8, 3, 9, 6, 9, 5, 4, 6, 1, 3, 8, 8, 8, 1, 4, 1, 0, 7, 8, 7, 8, 2, 2, 0, 1, 3, 0, 0, 7, 9, 4, 1, 3, 7, 5, 9, 4, 8, 8, 0

Offset: 1

David Cleaver, Mar 16 2025

Self-descriptive sequence: even indexed terms are the sequence itself, odd indexed terms are the decimal digits of the golden ratio.
This is an r1k1 fractal sequence, where r1k1 means: remove 1 term, keep 1 term, repeat.  The Removed terms are the sequence that has been fractalized, and the Kept terms are the original fractal sequence.
This fractal sequence is not a Kimberling fractal sequence because if you delete the first occurrence of each term, the remaining sequence is not the same as the original.

David Cleaver, Table of n, a(n) for n = 1..10000
Clark Kimberling, Fractal sequences.

Bisection gives A001622 (odd part).

Cf. A003602, A110766, A110779, A110812, A382128, A382129.

a(2n) = a(n); a(2n-1) = A001622(n), n >= 1.

a(n) = A001622(A003602(n)).

A110765 Let n in binary be a k-bit number with bits d_1, d_2, ..., d_k (concatenated). a(n) = 2^d_1 * 3^d_2 * ... * prime(k)^d_k, where prime(k) is the k-th prime.

Original entry on oeis.org

1, 2, 2, 6, 2, 10, 6, 30, 2, 14, 10, 70, 6, 42, 30, 210, 2, 22, 14, 154, 10, 110, 70, 770, 6, 66, 42, 462, 30, 330, 210, 2310, 2, 26, 22, 286, 14, 182, 154, 2002, 10, 130, 110, 1430, 70, 910, 770, 10010, 6, 78, 66, 858, 42, 546, 462, 6006, 30, 390, 330, 4290, 210, 2730

Offset: 0

Amarnath Murthy, Aug 12 2005

All terms after a(0) have 2-adic valuation equal to 1, i.e., they equal twice an odd (and squarefree) number, since the first digit in base two will always be "1". - M. F. Hasler, Mar 25 2011

2 appears at index n = 2^k for k >= 0, since such n_2 begins with "1" followed by k zeros, and 2^1 * 3^0 * ... * p_(k+1)^0 = 2. - Michael De Vlieger, Feb 28 2021

			n = 7: binary(7) = 111, and the first three primes are 2, 3, 5, so a(7) = 2^1 * 3^1 * 5^1 = 2*3*5 = 30.
n = 10: binary(10) = 1010, so a(10) = 2^1 * 3^0 * 5^1 * 7^0 = 2*1*5*1 = 10.

Peter Munn, Table of n, a(n) for n = 0..10000 (essentially from Reinhard Zumkeller)

Cf. A110766.
Cf. A030308, A000040, A000523, A019565.
Range of terms: {1} U A039956.

a110765 = product . zipWith (^) a000040_list .  reverse . a030308_row
-- Reinhard Zumkeller, Aug 28 2014

Array[Times @@ Prime@ Flatten@ Position[#, 1] &@ IntegerDigits[#, 2] &, 61] (* Michael De Vlieger, Feb 28 2021 *)

a(n)=factorback(Mat(vector(#n=binary(n),j,[prime(j),n[j]])~))

a(n)=prod(j=1,#n=binary(n),prime(j)^n[j])  \\ M. F. Hasler, Mar 25 2011

from sympy import prime
from operator import mul
from functools import reduce
def A110765(n):
    return reduce(mul, (prime(i) for i,d in enumerate(bin(n)[2:],start=1) if int(d)))
# Chai Wah Wu, Sep 05 2014

# implementation using recursion
from sympy import prime
def _A110765(n):
    nlen = len(n)
    return _A110765(n[:-1])*(prime(nlen) if int(n[-1]) else 1) if nlen > 1 else int(n) + 1
def A110765(n):
    return _A110765(bin(n)[2:])
# Chai Wah Wu, Sep 05 2014

a(0) = 1; a(2n) = a(n); a(2n+1) = a(n) * A000040(1+A000523(2n+1)), where A000040(k) is the k-th prime and A000523(k) = floor(log_2(k)) . - Peter Munn, Aug 26 2025

More terms from Stacy Hawthorne (shawtho1(AT)ashland.edu), Oct 31 2005
Name edited by Peter Munn, May 28 2025
a(0) prefixed by Peter Munn, Aug 26 2025

A110962 Fractalization of A025480, zero-based version of Kimberling's paraphrases sequence.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 3, 0, 0, 0, 4, 2, 2, 1, 5, 1, 1, 0, 6, 3, 3, 0, 7, 0, 0, 0, 8, 4, 4, 2, 9, 2, 2, 1, 10, 5, 5, 1, 11, 1, 1, 0, 12, 6, 6, 3, 13, 3, 3, 0, 14, 7, 7, 0, 15, 0, 0, 0, 16, 8, 8, 4, 17, 4, 4, 2, 18, 9, 9, 2, 19, 2, 2, 1, 20, 10, 10, 5, 21, 5, 5, 1, 22, 11, 11, 1, 23, 1, 1, 0, 24, 12, 12

Offset: 0

Alexandre Wajnberg, Sep 26 2005

Self-descriptive sequence: the terms at odd indices are the sequence itself, while the terms at even indices (the skeleton of this sequence) are the terms of A025480, which is a zero-based sequence of Kimberling's paraphrases sequence, A003602.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Clark Kimberling, Fractal sequences
Index entries for sequences related to binary expansion of n

Cf. A000265, A003602, A025480, A089309, A103391, A110812, A110779, A110766.

One less than A110963 (note also the different starting offsets).

A025480(n) = (n>>valuation(n*2+2, 2));
A110962(n) = if(!(n%2), A025480(n/2), A110962((n-1)/2)); \\ Antti Karttunen, Apr 18 2022

a(n) = n++; n>>=valuation(n, 2); n>>valuation(2*n+2, 2); \\ Ruud H.G. van Tol, Jun 23 2024

For even n, a(n) = A025480(n/2), for odd n, a(n) = a((n-1)/2). - Antti Karttunen, Apr 18 2022
a(2n+1) = a(4n+3) = a(n).
a(2n) = a(4n+1) = a(4n+2) = A025480(n/2).
a(4n) = a(8n+1) = a(8n+2) = n.
a(n) = A110963(1+n) - 1.

Entry edited and more terms added by Antti Karttunen, Apr 18 2022

A382128 Fractalization of the Recamán sequence.

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 6, 0, 2, 3, 7, 1, 13, 6, 20, 0, 12, 2, 21, 3, 11, 7, 22, 1, 10, 13, 23, 6, 9, 20, 24, 0, 8, 12, 25, 2, 43, 21, 62, 3, 42, 11, 63, 7, 41, 22, 18, 1, 42, 10, 17, 13, 43, 23, 16, 6, 44, 9, 15, 20, 45, 24, 14, 0, 46, 8, 79, 12, 113, 25, 78, 2, 114, 43, 77, 21, 39, 62, 78

Offset: 1

David Cleaver, Mar 16 2025

Self-descriptive sequence: even indexed terms are the sequence itself, odd indexed terms are the Recamán sequence.
This is an r1k1 fractal sequence, where r1k1 means: remove 1 term, keep 1 term, repeat.  The Removed terms are the sequence that has been fractalized, and the Kept terms are the original fractal sequence.
This fractal sequence is not a Kimberling fractal sequence because if you delete the first occurrence of each term, the remaining sequence is not the same as the original. This sequence fails to be a Kimberling fractal due to having consecutive terms that both appeared earlier in the sequence, starting with the 1 and 42 at index 48 and 49, respectively.

David Cleaver, Table of n, a(n) for n = 1..10000
Clark Kimberling, Fractal sequences.

Cf. A005132, A003602, A110766, A110779, A110812, A382129, A382130.

a(2n) = a(n); a(2n-1) = A005132(n), n >= 1.

a(n) = A005132(A003602(n)).

A382129 Fractalization of the prime numbers.

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 7, 2, 11, 5, 13, 3, 17, 7, 19, 2, 23, 11, 29, 5, 31, 13, 37, 3, 41, 17, 43, 7, 47, 19, 53, 2, 59, 23, 61, 11, 67, 29, 71, 5, 73, 31, 79, 13, 83, 37, 89, 3, 97, 41, 101, 17, 103, 43, 107, 7, 109, 47, 113, 19, 127, 53, 131, 2, 137, 59, 139, 23, 149, 61, 151, 11, 157

Offset: 1

David Cleaver, Mar 16 2025

Self-descriptive sequence: even indexed terms are the sequence itself, odd indexed terms are the prime numbers.
This is an r1k1 fractal sequence, where r1k1 means: remove 1 term, keep 1 term, repeat. The Removed terms are the sequence that has been fractalized, and the Kept terms are the original fractal sequence.
This fractal sequence is also a Kimberling fractal sequence because if you delete the first occurrence of each term, the remaining sequence is the same as the original.

David Cleaver, Table of n, a(n) for n = 1..10000
Clark Kimberling, Fractal sequences.

Cf. A000040, A003602, A110766, A110779, A110812, A382128, A382130.

a[n_] := Prime[(n/2^IntegerExponent[n, 2] + 1)/2]; Array[a, 100] (* Amiram Eldar, Mar 21 2025 *)

a(2n) = a(n); a(2n-1) = A000040(n), n >= 1.

a(n) = A000040(A003602(n)).

cp's OEIS Frontend

A110779 Fractalization of e.

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A110812 Fractalization of sqrt 2.

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A110963 Fractalization of Kimberling's paraphrases sequence beginning with 1.

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A382130 Fractalization of the golden ratio.

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A110765 Let n in binary be a k-bit number with bits d_1, d_2, ..., d_k (concatenated). a(n) = 2^d_1 * 3^d_2 * ... * prime(k)^d_k, where prime(k) is the k-th prime.

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A110962 Fractalization of A025480, zero-based version of Kimberling's paraphrases sequence.

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A382128 Fractalization of the Recamán sequence.

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A382129 Fractalization of the prime numbers.

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