A003602 -id:A003602 - cp's OEIS frontend

A269374 Permutation of natural numbers: a(1) = 1, a(n) = A255551(A001511(n), a(A003602(n))) - 1.

1, 2, 3, 6, 5, 4, 11, 8, 9, 10, 7, 18, 21, 28, 15, 12, 17, 22, 19, 38, 13, 16, 35, 26, 41, 58, 55, 102, 29, 40, 23, 14, 33, 46, 43, 80, 37, 52, 75, 56, 25, 34, 31, 60, 69, 100, 51, 44, 81, 118, 115, 206, 109, 160, 203, 152, 57, 82, 79, 144, 45, 64, 27, 20, 65, 94, 91, 164, 85, 124, 159, 120, 73, 106, 103, 186, 149, 220, 111, 96, 49

Offset: 1

Antti Karttunen, Mar 01 2016

Permutation obtained from the Lucky sieve.
This sequence can be represented as a binary tree. For n > 2, each left hand child is obtained by doubling the contents of the parent node and subtracting one, and each right hand child is obtained by applying A269372(n), when the parent node contains n:
                                    1
                                    |
                 ...................2...................
                3                                       6
      5......../ \........4                  11......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  9       10          7       18         21       28         15       12
17 22   19  38      13 16   35  26     41  58   55  102    29  40   23  14
etc.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Inverse: A269373.
Cf. A000035, A001511, A003602, A255551, A269372.
Cf. also A269375, A269377 and also A249814, A269384.

a(1) = 1, a(n) = A255551(A001511(n), a(A003602(n))) - 1.
a(1) = 1, a(2n) = A269372(a(n)), a(2n+1) = (2*a(n+1))-1.
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A269384 Permutation of natural numbers: a(1) = 1, a(n) = A255127(A001511(n), a(A003602(n))) - 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 18, 13, 20, 11, 10, 17, 26, 27, 34, 29, 44, 35, 30, 25, 38, 39, 48, 21, 32, 19, 12, 33, 50, 51, 64, 53, 80, 67, 58, 57, 86, 87, 108, 69, 104, 59, 54, 49, 74, 75, 94, 77, 116, 95, 84, 41, 62, 63, 78, 37, 56, 23, 16, 65, 98, 99, 124, 101, 152, 127, 112, 105, 158, 159, 198, 133, 200

Offset: 1

Antti Karttunen, Mar 01 2016

Permutation obtained from the Ludic sieve.
This sequence can be represented as a binary tree. For n > 2, each left hand child is obtained by doubling the contents of the parent node and subtracting one, and each right hand child is obtained by applying A269382(n), when the parent node contains n:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........8                   7......../ \........6
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  9       14         15       18         13       20         11       10
17 26   27  34     29  44   35  30     25  38   39  48     21  32   19  12
etc.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A269383.
Cf. A000035, A001511, A003602, A255127, A269382.
Cf. also A269385, A269387 and also A249814, A269374.

a(1) = 1, a(n) = A255127(A001511(n), a(A003602(n))) - 1.
a(1) = 1, a(2n) = A269382(a(n)), a(2n+1) = (2*a(n+1))-1.
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A347957 Dirichlet convolution of A001221 (omega) with A003602 (Kimberling's paraphrases).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 3, 6, 1, 9, 1, 7, 7, 4, 1, 14, 1, 11, 8, 9, 1, 13, 4, 10, 8, 13, 1, 28, 1, 5, 10, 12, 9, 25, 1, 13, 11, 16, 1, 34, 1, 17, 22, 15, 1, 17, 5, 25, 13, 19, 1, 38, 11, 19, 14, 18, 1, 49, 1, 19, 26, 6, 12, 46, 1, 23, 16, 44, 1, 36, 1, 22, 31, 25, 12, 52, 1, 21, 22, 24, 1, 60, 14, 25, 19, 25, 1, 86

Offset: 1

Antti Karttunen, Sep 20 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Jon Maiga, Computer-generated formulas for A347957, Sequence Machine.

Cf. A001221, A003602, A023900, A181988, A205745.

Cf. also A328203, A347954, A347955, A347956.

A003602(n) = (1+(n>>valuation(n,2)))/2;
A347957(n) = sumdiv(n,d,omega(n/d)*A003602(d));

a(n) = Sum_{d|n} A001221(n/d) * A003602(d).
From Antti Karttunen, Nov 13 2021: (Start)
The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. The first one is obvious, and even the second one should not be too hard to prove:
a(n) = Sum_{d|n} A023900(n/d) * A347956(d).
a(n) = Sum_{d|n} A181988(n/d) * A205745(d).
(End)

A349346 Dirichlet inverse of A181988, where A181988(n) = A001511(n)*A003602(n).

Original entry on oeis.org

1, -2, -2, 1, -3, 4, -4, 0, -1, 6, -6, -2, -7, 8, 4, 0, -9, 2, -10, -3, 5, 12, -12, 0, -4, 14, -2, -4, -15, -8, -16, 0, 7, 18, 6, -1, -19, 20, 8, 0, -21, -10, -22, -6, 3, 24, -24, 0, -9, 8, 10, -7, -27, 4, 8, 0, 11, 30, -30, 4, -31, 32, 4, 0, 9, -14, -34, -9, 13, -12, -36, 0, -37, 38, 8, -10, 9, -16, -40, 0, -4, 42

Offset: 1

Antti Karttunen, Nov 15 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001511, A003602, A181988, A349347.

Cf. also A347957, A349134, A349344.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA001511(n) = 1+valuation(n,2);
A003602(n) = (1+(n>>valuation(n,2)))/2;
A181988(n) = (A001511(n)*A003602(n));
v349346 = DirInverseCorrect(vector(up_to,n,A181988(n)));
A349346(n) = v349346[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A181988(n/d) * a(d).

a(n) = A349347(n) - A181988(n).

A349373 Dirichlet convolution of Kimberling's paraphrases (A003602) with Dirichlet inverse of Euler phi (A023900).

Original entry on oeis.org

1, 0, 0, -1, -1, 0, -2, -2, -1, 0, -4, 0, -5, 0, 2, -3, -7, 0, -8, 1, 3, 0, -10, 0, -3, 0, -2, 2, -13, 0, -14, -4, 5, 0, 8, 1, -17, 0, 6, 2, -19, 0, -20, 4, 5, 0, -22, 0, -5, 0, 8, 5, -25, 0, 14, 4, 9, 0, -28, -2, -29, 0, 8, -5, 17, 0, -32, 7, 11, 0, -34, 2, -35, 0, 4, 8, 23, 0, -38, 3, -3, 0, -40, -3, 23, 0, 14, 8

Offset: 1

Antti Karttunen, Nov 15 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003602, A023900.

Cf. A347954, A347955, A347956, A349136, A349370, A349371, A349372, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.

f[p_, e_] := (1 - p); d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] * d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A349373(n) = sumdiv(n,d,A003602(n/d)*A023900(d));

a(n) = Sum_{d|n} A003602(n/d) * A023900(d).

A351565 Odd part of Kimberling's paraphrases: a(n) = A000265(A003602(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 5, 3, 3, 1, 7, 1, 1, 1, 9, 5, 5, 3, 11, 3, 3, 1, 13, 7, 7, 1, 15, 1, 1, 1, 17, 9, 9, 5, 19, 5, 5, 3, 21, 11, 11, 3, 23, 3, 3, 1, 25, 13, 13, 7, 27, 7, 7, 1, 29, 15, 15, 1, 31, 1, 1, 1, 33, 17, 17, 9, 35, 9, 9, 5, 37, 19, 19, 5, 39, 5, 5, 3, 41, 21, 21, 11, 43, 11, 11, 3, 45, 23, 23, 3, 47

Offset: 1

Antti Karttunen, Mar 27 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A000265, A003602, A023758 (gives the positions of 1's after its initial zero-term).

Cf. also A336698, A336699.

A000265(n) = (n>>valuation(n,2));
A351565(n) = A000265(1+A000265(n));

def A351565(n): return (m:=1+(n>>(~n&n-1).bit_length()))>>(~m&m-1).bit_length()  # Chai Wah Wu, Jan 03 2024

a(n) = A000265(A003602(n)) = A000265(1+A000265(n)).

A353366 Dirichlet inverse of A110963, which is a fractalization of Kimberling's paraphrases sequence (A003602).

Original entry on oeis.org

1, -1, -1, 0, -2, 1, -1, 0, -2, 2, -2, 0, -4, 1, 3, 0, -5, 2, -3, 0, -4, 2, -2, 0, -3, 4, 1, 0, -8, -3, -1, 0, -5, 5, -1, 0, -10, 3, 5, 0, -11, 4, -6, 0, -4, 2, -2, 0, -12, 3, 3, 0, -14, -1, 4, 0, -9, 8, -8, 0, -16, 1, 14, 0, -1, 5, -9, 0, -14, 1, -5, 0, -19, 10, -4, 0, -16, -5, -3, 0, -12, 11, -11, 0, -2, 6, 10

Offset: 1

Antti Karttunen, Apr 18 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003602, A110963, A353367.

Cf. also A349134, A353368.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (1+(n>>valuation(n,2)))/2;
A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2));
v353366 = DirInverseCorrect(vector(up_to,n,A110963(n)));
A353366(n) = v353366[n];

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A353366(n): return 1 if n==1 else -sum(((1+(m:=d>>(~d&d-1).bit_length())>>(m+1&-m-1).bit_length())+1)*A353366(n//d) for d in divisors(n,generator=True) if d>1) # Chai Wah Wu, Jan 04 2024

a(1) = 1; a(n) = -Sum_{d|n, d < n} A110963(n/d) * a(d).

a(n) = A353367(n) - A110963(n).

A304100 a(n) = A003602(A048679(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 1, 5, 3, 2, 4, 1, 9, 5, 3, 6, 2, 7, 4, 1, 17, 9, 5, 10, 3, 11, 6, 2, 13, 7, 4, 8, 1, 33, 17, 9, 18, 5, 19, 10, 3, 21, 11, 6, 12, 2, 25, 13, 7, 14, 4, 15, 8, 1, 65, 33, 17, 34, 9, 35, 18, 5, 37, 19, 10, 20, 3, 41, 21, 11, 22, 6, 23, 12, 2, 49, 25, 13, 26, 7, 27, 14, 4, 29, 15, 8, 16, 1, 129, 65, 33, 66, 17, 67, 34, 9, 69, 35, 18, 36, 5, 73

Offset: 1

Antti Karttunen, May 13 2018

nonn

Positions of ones is given by the positive Fibonacci numbers: 1, 2, 3, 5, 8, 13, 21, ..., that is, A000045(n) from n >= 2 onward.
Positions of 2's is given by Lucas numbers larger than 3: 4, 7, 11, 18, ..., that is, A000032(n) from n >= 3 onward.
The restricted growth sequence transform of this sequence (almost certainly) is A003603.

Antti Karttunen, Table of n, a(n) for n = 1..17710

Cf. A000032, A000045, A003602, A003603, A035513, A048679, A304101.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A007814(n) = valuation(n,2);
A000265(n) = (n/2^valuation(n, 2));
A106151(n) = if(n<=1,n,if(n%2,1+(2*A106151((n-1)/2)),(2^(A007814(n)-1))*A106151(A000265(n))));
A048679(n) = if(!n,n,A106151(2*A003714(n)));
A003602(n) = (1+A000265(n))/2;
A304100(n) = A003602(A048679(n));

a(n) = A003602(A048679(n)).

For all i, j: a(i) = a(j) => A304101(i) = A304101(j).

A349137 a(n) = phi(A003602(n)), where A003602 is Kimberling's paraphrases, and phi is Euler totient function.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 6, 2, 4, 1, 6, 4, 4, 2, 10, 2, 4, 1, 12, 6, 6, 2, 8, 4, 8, 1, 16, 6, 6, 4, 18, 4, 8, 2, 12, 10, 10, 2, 22, 4, 8, 1, 20, 12, 12, 6, 18, 6, 12, 2, 28, 8, 8, 4, 30, 8, 16, 1, 20, 16, 16, 6, 24, 6, 12, 4, 36, 18, 18, 4, 24, 8, 16, 2, 40, 12, 12, 10, 42, 10, 20, 2, 24, 22, 22, 4, 46

Offset: 1

Antti Karttunen, Nov 13 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A000265, A003602, A349138 (inverse Möbius transform).

Cf. also A349136.

A003602(n) = (1+(n>>valuation(n,2)))/2;
A349137(n) = eulerphi(A003602(n));

a(n) = A000010(A003602(n)).

For all n >= 1, a(n) = a(2*n) = a(A000265(n)), a(2n-1) = A000010(n).

A349138 Inverse Möbius transform of A349137, where A349137(n) = phi(A003602(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 4, 6, 6, 3, 6, 7, 6, 8, 5, 7, 12, 5, 9, 14, 6, 5, 8, 15, 14, 12, 9, 9, 16, 9, 6, 20, 14, 11, 18, 19, 10, 16, 12, 13, 28, 11, 9, 34, 10, 9, 10, 23, 30, 20, 21, 19, 24, 17, 12, 34, 18, 9, 24, 31, 18, 34, 7, 29, 40, 17, 21, 30, 22, 13, 24, 37, 38, 38, 15, 29, 32, 17, 15, 52, 26, 13, 42, 51, 22, 30

Offset: 1

Antti Karttunen, Nov 13 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A003602, A349137.

Cf. also A349122.

A349137(n) = eulerphi((1+(n>>valuation(n,2)))/2);
A349138(n) = sumdiv(n,d,A349137(d));

a(n) = Sum_{d|n} A349137(d).

cp's OEIS Frontend

A269374 Permutation of natural numbers: a(1) = 1, a(n) = A255551(A001511(n), a(A003602(n))) - 1.

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A269384 Permutation of natural numbers: a(1) = 1, a(n) = A255127(A001511(n), a(A003602(n))) - 1.

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A347957 Dirichlet convolution of A001221 (omega) with A003602 (Kimberling's paraphrases).

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A349346 Dirichlet inverse of A181988, where A181988(n) = A001511(n)*A003602(n).

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A349373 Dirichlet convolution of Kimberling's paraphrases (A003602) with Dirichlet inverse of Euler phi (A023900).

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Mathematica

PARI

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A351565 Odd part of Kimberling's paraphrases: a(n) = A000265(A003602(n)).

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A353366 Dirichlet inverse of A110963, which is a fractalization of Kimberling's paraphrases sequence (A003602).

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A304100 a(n) = A003602(A048679(n)).

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A349137 a(n) = phi(A003602(n)), where A003602 is Kimberling's paraphrases, and phi is Euler totient function.

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