A003602 -id:A003602 - cp's OEIS frontend

A349136 Möbius transform of Kimberling's paraphrases, A003602.

1, 0, 1, 0, 2, 0, 3, 0, 3, 0, 5, 0, 6, 0, 4, 0, 8, 0, 9, 0, 6, 0, 11, 0, 10, 0, 9, 0, 14, 0, 15, 0, 10, 0, 12, 0, 18, 0, 12, 0, 20, 0, 21, 0, 12, 0, 23, 0, 21, 0, 16, 0, 26, 0, 20, 0, 18, 0, 29, 0, 30, 0, 18, 0, 24, 0, 33, 0, 22, 0, 35, 0, 36, 0, 20, 0, 30, 0, 39, 0, 27, 0, 41, 0, 32, 0, 28, 0, 44, 0, 36, 0, 30, 0, 36

Offset: 1

Antti Karttunen, Nov 13 2021

nonn

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

Agrees with A055034 on odd arguments.
Cf. A000010, A003602, A008683, A349134.
Cf. A000004, A072451 (even and odd bisection).
Cf. also A347233, A349127, A349137.

with(numtheory): a:=proc(n) if n=1 then 1; elif n mod 2 = 0 then 0; else phi(n)/2; fi: end proc: seq(a(n), n=1..60); # Ridouane Oudra, Jul 13 2023

k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, MoebiusMu[#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

A349136(n) = if(1==n,1, if(n%2, eulerphi(n)/2, 0));

A003602(n) = (1+(n>>valuation(n,2)))/2;
A349136(n) = sumdiv(n,d,moebius(d)*A003602(n/d));

from sympy import totient
def A349136(n): return totient(n)+1>>1 if n&1 else 0 # Chai Wah Wu, Nov 24 2023

a(n) = Sum_{d|n} A008683(d) * A003602(n/d).
a(1) = 1, a(n) = A000010(n)/2 for odd n > 1, a(n) = 0 for even n.
For all n >= 1, a(2*n-1) = A055034(2*n-1) = A072451(n).
a(n) = phi(n) - (1/2)*phi(2n), for n>1. - Ridouane Oudra, Jul 13 2023
Sum_{k=1..n} a(k) ~ (1/Pi^2)*n^2. - Amiram Eldar, Jul 15 2023

A349134 Dirichlet inverse of Kimberling's paraphrases, A003602.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 13 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003602, A349135, A349136.

Cf. also A323881, A349125.

k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#]*k[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

up_to = 16384;
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;
v349134 = DirInverseCorrect(vector(up_to,n,A003602(n)));
A349134(n) = v349134[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A003602(n/d) * a(d).

a(n) = A349135(n) - A003602(n).

A349371 Inverse Möbius transform of Kimberling's paraphrases (A003602).

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 5, 4, 8, 8, 7, 9, 8, 10, 14, 5, 10, 16, 11, 12, 18, 14, 13, 12, 17, 16, 22, 15, 16, 28, 17, 6, 26, 20, 26, 24, 20, 22, 30, 16, 22, 36, 23, 21, 42, 26, 25, 15, 30, 34, 38, 24, 28, 44, 38, 20, 42, 32, 31, 42, 32, 34, 55, 7, 44, 52, 35, 30, 50, 52, 37, 32, 38, 40, 65, 33, 50, 60, 41, 20, 63, 44, 43

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

Dirichlet convolution of sigma (A000203) with A349431, or equally, A264740 with A349447. - Antti Karttunen, Nov 21 2021

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

Cf. A000203, A264740.
Cf. also A347954, A347955, A347956, A349136, A349370, A349372, A349373, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.
Cf. also A349393.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A349371(n) = sumdiv(n,d,A003602(d));

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

a(n) = Sum_{d|n} A000203(n/d)*A349431(d) = Sum_{d|n} A264740(n/d)*A349447(d). - Antti Karttunen, Nov 21 2021

A349431 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, -1, -1, -1, -2, 1, -3, -1, -1, 2, -5, 1, -6, 3, 4, -1, -8, 1, -9, 2, 6, 5, -11, 1, -2, 6, -1, 3, -14, -4, -15, -1, 10, 8, 12, 1, -18, 9, 12, 2, -20, -6, -21, 5, 4, 11, -23, 1, -3, 2, 16, 6, -26, 1, 20, 3, 18, 14, -29, -4, -30, 15, 6, -1, 24, -10, -33, 8, 22, -12, -35, 1, -36, 18, 4, 9, 30, -12, -39, 2, -1, 20

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Dirichlet convolution of this sequence with A000010 gives A349136, which also proves the formula involving A023900.

Convolution with A000203 gives A349371.

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

Sequence A297381 negated.
Cf. A003602, A023900, A055615, A297381, A349432 (Dirichlet inverse), A349433 (sum with it).
Cf. also A000010, A000203, A349136, A349371, and also A349444, A349447.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, # * MoebiusMu [#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A055615(n) = (n*moebius(n));
A349431(n) = sumdiv(n,d,A003602(n/d)*A055615(d));

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A349431(n) = if(!bitand(n,n-1),A023900(n),A023900(n)/2);

a(n) = Sum_{d|n} A003602(n/d) * A055615(d).
a(n) = A023900(n) when n is a power of 2, and a(n) = A023900(n)/2 for all other numbers.
a(n) = -A297381(n).

A347954 Dirichlet convolution of A003602 with A342001.

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 6, 4, 11, 1, 20, 1, 14, 13, 10, 1, 26, 1, 29, 16, 20, 1, 37, 5, 23, 12, 38, 1, 81, 1, 15, 22, 29, 19, 62, 1, 32, 25, 55, 1, 106, 1, 56, 48, 38, 1, 59, 6, 48, 31, 65, 1, 74, 25, 73, 34, 47, 1, 191, 1, 50, 61, 21, 28, 156, 1, 83, 40, 151, 1, 112, 1, 59, 60, 92, 28, 181, 1, 89, 34, 65, 1, 254, 34

Offset: 1

Antti Karttunen, Sep 20 2021

nonn

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

Cf. A003415, A003557, A003602, A342001.

Cf. also A347234, A347955, A347956.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003602(n) = (1+(n>>valuation(n,2)))/2;
A342001(n) = (A003415(n) / A003557(n));
A347954(n) = sumdiv(n,d,A003602(d)*A342001(n/d));

a(n) = Sum_{d|n} A003602(d) * A342001(n/d).

A349390 Dirichlet convolution of A126760 with Kimberling's paraphrases, A003602.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 4, 8, 10, 10, 9, 12, 14, 17, 5, 15, 16, 17, 15, 24, 20, 20, 12, 28, 24, 22, 21, 25, 34, 27, 6, 35, 30, 47, 24, 32, 34, 42, 20, 35, 48, 37, 30, 50, 40, 40, 15, 54, 56, 53, 36, 45, 44, 71, 28, 60, 50, 50, 51, 52, 54, 71, 7, 84, 70, 57, 45, 71, 94, 60, 32, 62, 64, 100, 51, 99, 84, 67, 25, 63, 70, 70

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A003602, A126760.

Cf. A347233, A347234, A349391, A349392, A349393, A349395, A349431, A349444, A349447 for other Dirichlet convolutions of A126760. And also A349370.

f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, f[#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349390(n) = sumdiv(n,d,A126760(n/d)*A003602(d));

a(n) = Sum_{d|n} A126760(n/d) * A003602(d).

A349444 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A092673 (Dirichlet inverse of A001511).

Original entry on oeis.org

1, -1, 1, 0, 2, -1, 3, 0, 3, -2, 5, 0, 6, -3, 4, 0, 8, -3, 9, 0, 6, -5, 11, 0, 10, -6, 9, 0, 14, -4, 15, 0, 10, -8, 12, 0, 18, -9, 12, 0, 20, -6, 21, 0, 12, -11, 23, 0, 21, -10, 16, 0, 26, -9, 20, 0, 18, -14, 29, 0, 30, -15, 18, 0, 24, -10, 33, 0, 22, -12, 35, 0, 36, -18, 20, 0, 30, -12, 39, 0, 27, -20, 41, 0, 32

Offset: 1

Antti Karttunen, Nov 18 2021

sign

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

Cf. A001511, A003602, A008683, A092673, A349445 (Dirichlet inverse), A349446 (sum with it).

Cf. also A349431, A349447.

s[n_] := MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]]; k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, s[#]*k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A092673(n) = if(n<1, 0, moebius(n) - if( n%2, 0, moebius(n/2))); \\ From A092673
A349444(n) = sumdiv(n,d,A003602(n/d)*A092673(d));

a(n) = Sum_{d|n} A003602(n/d) * A092673(d).

A249814 "Mountains of Eratosthenes" permutation: a(1) = 1, a(n) = A249741(A001511(n), a(A003602(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 24, 13, 20, 11, 10, 17, 26, 27, 34, 29, 44, 47, 48, 25, 38, 39, 54, 21, 32, 19, 12, 33, 50, 51, 64, 53, 80, 67, 76, 57, 86, 87, 114, 93, 140, 95, 120, 49, 74, 75, 94, 77, 116, 107, 90, 41, 62, 63, 84, 37, 56, 23, 16, 65, 98, 99, 124, 101, 152, 127, 118, 105, 158, 159, 204, 133, 200, 151, 142

Offset: 1

Antti Karttunen, Nov 06 2014

This sequence is a "recursed variant" of A249811.
From Antti Karttunen, Jan 18 2015: (Start)
This can be viewed as an entanglement or encoding permutation where the complementary pairs of sequences to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with another complementary pair: even numbers in the order they appear in A253886 and odd numbers in their usual order: (A253886/A005408).
From the above follows also that this sequence can be represented as a binary tree. Each child to the left is obtained by doubling the parent and subtracting one, and each child to the right is obtained by applying A253886 to the parent:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........8                   7......../ \........6
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  9       14         15       24         13       20         11       10
17 26   27  34     29  44   47  48     25  38   39  54     21  32   19  12
(End)
For listening I recommend some (mostly) percussive MIDI-instrument and the pitch offset set to at least 29 and the tempo (rate) to about 60. - Antti Karttunen, Feb 17 2015

Antti Karttunen, Table of n, a(n) for n = 1..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A249813.
Similar or related permutations: A246684, A249811, A250244, A252755.
Cf. A000079, A000265, A001511, A003602, A006093, A083140, A083221, A249741, A250469, A253886.
Compare also the scatterplot of this sequence to the graphs of A252755 and A246684.
Differs from A246684 for the first time at n=14, where a(14) = 20, while A246684(14) = 26.

In the following formulas, A083221 and A249741 are interpreted as bivariate functions:
a(1) = 1, for n>1: a(n) = A083221(A001511(n), a(A003602(n))) - 1 = A249741(A001511(n), a(A003602(n))).
a(1) = 1, a(2n) = A253886(a(n)), a(2n+1) = (2*a(n+1))-1. - Antti Karttunen, Jan 18 2015
As a composition of other permutations:
a(n) = A250244(A246684(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A000079(n-1)) = A006093(n).

A347955 Dirichlet convolution of A003415 (arithmetic derivative) with A003602 (Kimberling's paraphrases).

Original entry on oeis.org

0, 1, 1, 5, 1, 8, 1, 17, 8, 11, 1, 32, 1, 14, 13, 49, 1, 44, 1, 47, 16, 20, 1, 100, 13, 23, 44, 62, 1, 81, 1, 129, 22, 29, 19, 156, 1, 32, 25, 151, 1, 106, 1, 92, 86, 38, 1, 276, 18, 92, 31, 107, 1, 206, 25, 202, 34, 47, 1, 301, 1, 50, 111, 321, 28, 156, 1, 137, 40, 151, 1, 460, 1, 59, 120, 152, 28, 181, 1, 423, 206

Offset: 1

Antti Karttunen, Sep 20 2021

nonn

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

Cf. A003415, A003602.

Cf. also A347954, A347956.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003602(n) = (1+(n>>valuation(n,2)))/2;
A347955(n) = sumdiv(n,d,A003415(n/d)*A003602(d));

a(n) = Sum_{d|n} A003415(n/d) * A003602(d).

A349447 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A326937 (Dirichlet inverse of A000265).

Original entry on oeis.org

1, 0, -1, 0, -2, 0, -3, 0, -1, 0, -5, 0, -6, 0, 4, 0, -8, 0, -9, 0, 6, 0, -11, 0, -2, 0, -1, 0, -14, 0, -15, 0, 10, 0, 12, 0, -18, 0, 12, 0, -20, 0, -21, 0, 4, 0, -23, 0, -3, 0, 16, 0, -26, 0, 20, 0, 18, 0, -29, 0, -30, 0, 6, 0, 24, 0, -33, 0, 22, 0, -35, 0, -36, 0, 4, 0, 30, 0, -39, 0, -1, 0, -41, 0, 32, 0, 28

Offset: 1

Antti Karttunen, Nov 19 2021

sign

Dirichlet convolution of this sequence with A264740 is A349371.

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

Cf. A000265, A003602, A326937, A349448 (Dirichlet inverse).

Cf. also A264740, A349371, A349431, A349444.

k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, MoebiusMu[#] * # / 2^IntegerExponent[#, 2] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A003602(n) = (1+(n>>valuation(n,2)))/2;
A006519(n) = (1<A055615(n) = (n*moebius(n));
A326937(n) = (A055615(n)/A006519(n));
A349447(n) = sumdiv(n,d,A003602(d)*A326937(n/d));

a(n) = Sum_{d|n} A003602(d) * A326937(n/d).

cp's OEIS Frontend

A349136 Möbius transform of Kimberling's paraphrases, A003602.

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A349134 Dirichlet inverse of Kimberling's paraphrases, A003602.

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A349371 Inverse Möbius transform of Kimberling's paraphrases (A003602).

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A349431 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A055615 (Dirichlet inverse of n).

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A347954 Dirichlet convolution of A003602 with A342001.

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A349390 Dirichlet convolution of A126760 with Kimberling's paraphrases, A003602.

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A349444 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A092673 (Dirichlet inverse of A001511).

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A249814 "Mountains of Eratosthenes" permutation: a(1) = 1, a(n) = A249741(A001511(n), a(A003602(n))).

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