A347233 -id:A347233 - cp's OEIS frontend

A126760 a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.

0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 5, 3, 2, 1, 6, 1, 7, 2, 3, 4, 8, 1, 9, 5, 1, 3, 10, 2, 11, 1, 4, 6, 12, 1, 13, 7, 5, 2, 14, 3, 15, 4, 2, 8, 16, 1, 17, 9, 6, 5, 18, 1, 19, 3, 7, 10, 20, 2, 21, 11, 3, 1, 22, 4, 23, 6, 8, 12, 24, 1, 25, 13, 9, 7, 26, 5, 27, 2, 1, 14, 28, 3, 29, 15, 10, 4, 30, 2

Offset: 0

N. J. A. Sloane, Feb 19 2007

nonn

For further information see A126759, which provided the original motivation for this sequence.
From Antti Karttunen, Jan 28 2015: (Start)
The odd bisection of the sequence gives A253887, and the even bisection gives the sequence itself.
A254048 gives the sequence obtained when this sequence is restricted to A007494 (numbers congruent to 0 or 2 mod 3).
For all odd numbers k present in square array A135765, a(k) = the column index of k in that array. (End)
A322026 and this sequence (without the initial zero) are ordinal transforms of each other. - Antti Karttunen, Feb 09 2019
Also ordinal transform of A065331 (after the initial 0). - Antti Karttunen, Sep 08 2024

Antti Karttunen, Table of n, a(n) for n = 0..19683

One less than A126759.
Cf. A003586, A003602, A007310, A064989, A249746, A253887, A254048, A273669, A322317, A323881 (Dirichlet inverse), A323882.
Cf. A347233 (Möbius transform) and also A349390, A349393, A349395 for other Dirichlet convolutions.
Ordinal transform of A065331 and of A322026 (after the initial 0).
Related arrays: A135765, A254102.

f[n_] := Block[{a}, a[0] = 0; a[1] = a[2] = a[3] = 1; a[x_] := Which[EvenQ@ x, a[x/2], Mod[x, 3] == 0, a[x/3], Mod[x, 6] == 1, 2 (x - 1)/6 + 1, Mod[x, 6] == 5, 2 (x - 5)/6 + 2]; Table[a@ i, {i, 0, n}]] (* Michael De Vlieger, Feb 03 2015 *)

A126760(n)={n&&n\=3^valuation(n,3)<M. F. Hasler, Jan 19 2016

a(n) = A126759(n)-1. [The original definition.]
From Antti Karttunen, Jan 28 2015: (Start)
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.
Or with the last clause represented in another way:
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n-1) = 2n.
Other identities. For all n >= 1:
a(n) = A253887(A003602(n)).
a(6n-3) = a(4n-2) = a(2n-1) = A253887(n).
(End)
a(n) = A249746(A003602(A064989(n))). - Antti Karttunen, Feb 04 2015
a(n) = A323882(4*n). - Antti Karttunen, Apr 18 2022

Name replaced with an independent recurrence and the old description moved to the Formula section - Antti Karttunen, Jan 28 2015

A349136 Möbius transform of Kimberling's paraphrases, A003602.

Original entry on oeis.org

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

A347234 Dirichlet convolution of A126760 with A342001.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 6, 3, 10, 1, 17, 1, 13, 11, 10, 1, 16, 1, 26, 14, 18, 1, 31, 4, 21, 6, 35, 1, 61, 1, 15, 19, 26, 17, 36, 1, 29, 22, 49, 1, 82, 1, 50, 28, 34, 1, 49, 5, 36, 27, 59, 1, 28, 22, 67, 30, 42, 1, 139, 1, 45, 37, 21, 25, 117, 1, 74, 35, 127, 1, 63, 1, 53, 40, 83, 25, 138, 1, 79, 10, 58, 1, 190, 30, 61

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

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

Cf. A003415, A003557, A126760, A342001.

Cf. also A346485, A347233, A347235, A347395.

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]));
A342001(n) = (A003415(n) / A003557(n));
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A347234(n) = sumdiv(n,d,A126760(d)*A342001(n/d));

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

A349393 Inverse Möbius transform of A126760.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 3, 6, 5, 6, 6, 8, 6, 5, 7, 6, 8, 9, 8, 10, 9, 8, 12, 12, 4, 12, 11, 12, 12, 6, 10, 14, 18, 9, 14, 16, 12, 12, 15, 16, 16, 15, 9, 18, 17, 10, 21, 24, 14, 18, 19, 8, 26, 16, 16, 22, 21, 18, 22, 24, 12, 7, 30, 20, 24, 21, 18, 36, 25, 12, 26, 28, 24, 24, 34, 24, 28, 15, 5, 30, 29, 24, 38, 32, 22

Offset: 1

Antti Karttunen, Nov 16 2021

nonn

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

Cf. A347233, A347234, A349390, A349391, A349392, A349395 for other Dirichlet convolutions of A126760. And also A349371.

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

A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349393(n) = sumdiv(n,d,A126760(d));

a(n)=my(a=valuation(n,2),b=valuation(n,3),c=(a+1)*(b+1)); sumdiv(n/3^b>>a,d, d\6*2+d%3)*c; \\ Charles R Greathouse IV, Nov 16 2021

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

A347235 Dirichlet convolution of Euler phi with A342001, where A342001(n) = A003415(n) / A003557(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 7, 4, 12, 1, 21, 1, 16, 14, 15, 1, 27, 1, 33, 18, 24, 1, 47, 6, 28, 13, 45, 1, 87, 1, 31, 26, 36, 22, 69, 1, 40, 30, 75, 1, 119, 1, 69, 51, 48, 1, 99, 8, 63, 38, 81, 1, 84, 30, 103, 42, 60, 1, 219, 1, 64, 67, 63, 34, 183, 1, 105, 50, 183, 1, 153, 1, 76, 75, 117, 34, 215, 1, 159, 40, 84, 1, 303, 42

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

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

Cf. A000010, A003415, A003557, A342001.

Cf. also A346485, A347131, A347233, A347234, A347395.

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]));
A342001(n) = (A003415(n) / A003557(n));
A347235(n) = sumdiv(n,d,eulerphi(d)*A342001(n/d));

A347235(n) = sum(k=1,n,A342001(gcd(n,k))); \\ (Slow) - Antti Karttunen, Sep 02 2021

a(n) = Sum_{d|n} A000010(n/d) * A342001(d).

a(n) = Sum_{k=1..n} A342001(gcd(n,k)). - Antti Karttunen, Sep 02 2021

A349395 Dirichlet convolution of A126760 with Liouville's lambda.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 4, 0, 0, 1, 5, 0, 6, 1, 0, 0, 7, 0, 8, 0, 0, 2, 9, 0, 10, 0, 0, 0, 8, 1, 12, 0, 0, 0, 13, 0, 14, 3, 1, 0, 15, 0, 15, 0, 0, 4, 17, 0, 14, 0, 0, 0, 19, 0, 20, 0, 2, 1, 16, 0, 22, 5, 0, 0, 23, 0, 24, 0, 0, 6, 20, 0, 26, 1, 1, 0, 27, 0, 22, 0, 0, 0, 29, 0, 24, 7, 0, 0, 24, 0, 32, 0

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A008836, A126760.

Cf. A347233, A347234, A349390, A349391, A349392, A349393 for other Dirichlet convolutions of A126760. And also A349375.

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

A008836(n) = ((-1)^bigomega(n));
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349395(n) = sumdiv(n,d,A126760(n/d)*A008836(d));

A349392 Dirichlet convolution of A126760 with tau (number of divisors function).

Original entry on oeis.org

1, 3, 3, 6, 4, 9, 5, 10, 6, 12, 6, 18, 7, 15, 12, 15, 8, 18, 9, 24, 15, 18, 10, 30, 16, 21, 10, 30, 12, 36, 13, 21, 18, 24, 26, 36, 15, 27, 21, 40, 16, 45, 17, 36, 24, 30, 18, 45, 26, 48, 24, 42, 20, 30, 35, 50, 27, 36, 22, 72, 23, 39, 30, 28, 40, 54, 25, 48, 30, 78, 26, 60, 27, 45, 48, 54, 44, 63, 29, 60, 15, 48

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A000005, A126760.

Cf. A347233, A347234, A349390, A349391, A349393, A349395 for other Dirichlet convolutions of A126760. And also A349372.

f[n_] := 2 * Floor[(m = n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3])/6] + Mod[m, 3]; a[n_] := DivisorSum[n, f[#] * DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349392(n) = sumdiv(n,d,A126760(n/d)*numdiv(d));

a(n) = Sum_{d|n} A126760(n/d) * A000005(d).

A349391 Dirichlet convolution of A126760 with omega.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 5, 1, 7, 1, 6, 5, 4, 1, 7, 1, 9, 6, 7, 1, 10, 3, 8, 3, 11, 1, 16, 1, 5, 7, 9, 7, 12, 1, 10, 8, 13, 1, 20, 1, 13, 9, 11, 1, 13, 4, 18, 9, 15, 1, 10, 8, 16, 10, 13, 1, 27, 1, 14, 11, 6, 9, 24, 1, 17, 11, 32, 1, 17, 1, 16, 18, 19, 9, 28, 1, 17, 4, 17, 1, 34, 10, 18, 13, 19, 1, 27, 10, 21

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

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

Cf. A001221, A126760.

Cf. A347233, A347234, A349390, A349392, A349393, A349395 for other Dirichlet convolutions of A126760. And also A347957.

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

A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A349391(n) = sumdiv(n,d,A126760(n/d)*omega(d));

a(n) = Sum_{d|n} A126760(n/d) * A001221(d).

A349339 Odd bisection of the Möbius transform of A126760.

Original entry on oeis.org

1, 0, 1, 2, 0, 3, 4, 0, 5, 6, 0, 7, 7, 0, 9, 10, 0, 8, 12, 0, 13, 14, 0, 15, 14, 0, 17, 14, 0, 19, 20, 0, 16, 22, 0, 23, 24, 0, 20, 26, 0, 27, 22, 0, 29, 24, 0, 24, 32, 0, 33, 34, 0, 35, 36, 0, 37, 30, 0, 32, 37, 0, 33, 42, 0, 43, 36, 0, 45, 46, 0, 40, 38, 0, 49, 50, 0, 40, 52, 0, 44, 54, 0, 55, 52, 0, 57, 40, 0, 59

Offset: 1

Antti Karttunen, Nov 16 2021

nonn

Cf. A126760, A347233.

Cf. also A072451 (the odd bisection of the Möbius transform of A003602).

A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A347233(n) = sumdiv(n,d,moebius(n/d)*A126760(d));
A349339(n) = A347233(n+n-1);

a(n) = A347233(2*n-1).

cp's OEIS Frontend

A126760 a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.

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A349136 Möbius transform of Kimberling's paraphrases, A003602.

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A347234 Dirichlet convolution of A126760 with A342001.

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

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A349393 Inverse Möbius transform of A126760.

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A347235 Dirichlet convolution of Euler phi with A342001, where A342001(n) = A003415(n) / A003557(n).

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A349395 Dirichlet convolution of A126760 with Liouville's lambda.

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A349392 Dirichlet convolution of A126760 with tau (number of divisors function).

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