A347395 -id:A347395 - cp's OEIS frontend

A347396 a(n) = A347395(A276086(n)), where A347395 is Dirichlet convolution of Liouville's lambda with A342001.

0, 1, 1, 3, 1, 2, 1, 5, 6, 14, 5, 9, 1, 2, 3, 5, 2, 3, 2, 6, 8, 16, 6, 10, 2, 3, 5, 7, 3, 4, 1, 7, 8, 20, 7, 13, 10, 34, 44, 92, 34, 58, 7, 13, 20, 32, 13, 19, 16, 40, 56, 104, 40, 64, 13, 19, 32, 44, 19, 25, 1, 2, 3, 5, 2, 3, 5, 9, 14, 22, 9, 13, 2, 3, 5, 7, 3, 4, 6, 10, 16, 24, 10, 14, 3, 4, 7, 9, 4, 5, 2, 8, 10, 22

Offset: 0

Antti Karttunen, Sep 02 2021

The scatter plot looks quite peculiar. - Antti Karttunen, Sep 20 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A003557, A008836, A276086, A342001, A347395.

Cf. also A342002, A346471, A347126, A347232, A347962.

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));
A008836(n) = ((-1)^bigomega(n));
A347395(n) = sumdiv(n,d,A008836(n/d)*A342001(d));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A347396(n) = A347395(A276086(n));

A366803 Dirichlet convolution of Liouville's lambda with A342001 applied to Doudna sequence: a(n) = A347395(A005940(1+n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 2, 1, 5, 6, 3, 1, 2, 2, 2, 1, 7, 8, 5, 10, 14, 5, 5, 1, 2, 3, 2, 2, 4, 2, 3, 1, 11, 12, 7, 14, 20, 7, 9, 16, 34, 44, 14, 7, 9, 10, 5, 1, 2, 3, 2, 5, 5, 2, 3, 2, 6, 8, 4, 2, 3, 3, 3, 1, 13, 14, 11, 16, 32, 11, 13, 18, 54, 68, 20, 11, 13, 14, 9, 22, 76, 92, 34, 124, 92, 34, 22, 11, 13, 20, 9, 16

Offset: 0

Antti Karttunen, Oct 26 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A003415, A005940, A008836, A342001, A347395, A366805 (rgs-transform).

Cf. also A347396, A366795, A366801.

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]));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A342001(n) = (A003415(n) / A003557(n));
A008836(n) = ((-1)^bigomega(n));
A347395(n) = sumdiv(n,d,A008836(n/d)*A342001(d));
A366803(n) = A347395(A005940(1+n));

A342001 Arithmetic derivative without its inherited divisor; the arithmetic derivative of n divided by A003557(n), which is a common divisor of both n and A003415(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 2, 7, 1, 8, 1, 9, 8, 4, 1, 7, 1, 12, 10, 13, 1, 11, 2, 15, 3, 16, 1, 31, 1, 5, 14, 19, 12, 10, 1, 21, 16, 17, 1, 41, 1, 24, 13, 25, 1, 14, 2, 9, 20, 28, 1, 9, 16, 23, 22, 31, 1, 46, 1, 33, 17, 6, 18, 61, 1, 36, 26, 59, 1, 13, 1, 39, 11, 40, 18, 71, 1, 22, 4, 43, 1, 62, 22, 45, 32, 35, 1, 41, 20

Offset: 1

Antti Karttunen, Feb 28 2021

nonn

See also the scatter plot of A342002 that seems to reveal some interesting internal structure in this sequence, not fully explained by the regularity of primorial base expansion used in the latter sequence. - Antti Karttunen, May 09 2022

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, A341998, A342416, A342459.
Cf. A342002 [= a(A276086(n))], A342463 [= a(A342456(n))], A351945 [= a(A181819(n))], A353571 [= a(A003961(n))].
Cf. A346485 (Möbius transform), A347395 (convolution with Liouville's lambda), A347961 (with itself), and A347234, A347235, A347954, A347959, A347963, A349396, A349612 (for convolutions with other sequences).
Cf. A007947.

Array[#1/#2 & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], #/Times @@ FactorInteger[#][[All, 1]]} &, 91] (* Michael De Vlieger, Mar 11 2021 *)

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));

from math import prod
from sympy import factorint
def A342001(n):
    q = prod(f:=factorint(n))
    return sum(q*e//p for p, e in f.items()) # Chai Wah Wu, Nov 04 2022

a(n) = A003415(n) / A003557(n).
For all n >= 0, a(A276086(n)) = A342002(n).
a(n) = A342414(n) * A342416(n) = A342459(n) * A342919(n). - Antti Karttunen, Apr 30 2022
Dirichlet g.f.: Dirichlet g.f. of A007947 * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)) = zeta(s) * Product_{p prime} (1+p^(1-s)-p^(-s)) * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 05 2022
Sum_{k=1..n} a(k) ~ c * A065464 * Pi^2 * n^2 / 12, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, May 09 2022

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).

A346485 Möbius transform of A342001, where A342001(n) = A003415(n)/A003557(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 2, 1, 7, 6, 1, 1, 1, 1, 4, 8, 11, 1, 2, 1, 13, 1, 6, 1, 14, 1, 1, 12, 17, 10, 0, 1, 19, 14, 4, 1, 20, 1, 10, 4, 23, 1, 2, 1, 1, 18, 12, 1, 1, 14, 6, 20, 29, 1, 8, 1, 31, 6, 1, 16, 32, 1, 16, 24, 34, 1, 0, 1, 37, 2, 18, 16, 38, 1, 4, 1, 41, 1, 12, 20, 43, 30, 10, 1, 4, 18, 22, 32, 47

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

Conjecture 1: After the initial zero, the positions of other zeros is given by A036785.

Conjecture 2: No negative terms. Checked up to n = 2^24.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A008683, A036785, A342001, A347232 [= a(A276086(n))].

Cf. also A300251, A300717, A347234, 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));
A346485(n) = sumdiv(n,d,moebius(n/d)*A342001(d));

a(n) = Sum_{d|n} A008683(n/d) * A342001(d).
Dirichlet g.f.: Product_{p prime} (1+p^(1-s)-p^(-s)) * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 08 2022
Sum_{k=1..n} a(k) ~ c * A065464 * n^2 / 2, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, Mar 04 2023

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

A369069 Dirichlet convolution of Liouville's lambda (A008836) with A083345, where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 1, 3, 1, 5, 1, 0, 1, 7, 6, -1, 1, 2, 1, 0, 8, 11, 1, 8, 1, 13, 0, 0, 1, 14, 1, 6, 12, 17, 10, 0, 1, 19, 14, 14, 1, 20, 1, 0, 5, 23, 1, -3, 1, 2, 18, 0, 1, 0, 14, 20, 20, 29, 1, 0, 1, 31, 7, -3, 16, 32, 1, 0, 24, 34, 1, 5, 1, 37, 3, 0, 16, 38, 1, -5, 4, 41, 1, 0, 20, 43, 30, 32, 1, 9, 18, 0, 32, 47

Offset: 1

Antti Karttunen, Jan 16 2024

sign

In contrast to A347395, this sequence has also nonpositive values after the initial term. It seems that A342090 gives most of the positions of nonpositive terms here, apart from k = 0, 729, 1458, 3645, 5103, 5832, 7290, ..., etc.

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

Cf. A003415, A008836, A083345, A342090.

Cf. also A347395, A369068.

A008836(n) = ((-1)^bigomega(n));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369069(n) = sumdiv(n,d,A008836(n/d)*A083345(d));

a(n) = Sum_{d|n} A008836(n/d) * A083345(d).

A349396 Dirichlet convolution of A342001 ({arithmetic derivative of n}/A003557(n)) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, -1, -1, 0, 1, -2, 1, 0, 0, -2, 1, -6, 1, -2, 0, 0, 1, -2, -3, 0, -3, -2, 1, 0, 1, -3, 0, 0, 0, 2, 1, 0, 0, -2, 1, 0, 1, -2, -6, 0, 1, -2, -5, -20, 0, -2, 1, -6, 0, -2, 0, 0, 1, 0, 1, 0, -6, -4, 0, 0, 1, -2, 0, 0, 1, 8, 1, 0, -20, -2, 0, 0, 1, -2, -5, 0, 1, 0, 0, 0, 0, -2, 1, 0, 0, -2, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 18 2021

sign

Dirichlet convolution of this sequence with A000010 (Euler phi) is A346485.

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

Cf. A003415, A003557, A055615, A342001.
Cf. A346485, A347234,  A347235, A347395, A347954, A347959, A347961, A347963 for Dirichlet convolutions of A342001 with other sequences.
Cf. also A349394.

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));
A055615(n) = (n*moebius(n));
A349396(n) = sumdiv(n,d,A342001(n/d)*A055615(d));

a(n) = Sum_{d|n} A055615(d) * A342001(n/d).

A366805 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366803(i) = A366803(j) for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 26 2023

Restricted growth sequence transform of A366803.

The scatter plot has quite interesting structure.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A005940, A008836, A347395, A366803.

Cf. also A366796, A366802.

\\ Needs also program from A366803:
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v366805 = rgs_transform(vector(1+up_to,n,A366803(n-1)));
A366805(n) = v366805[1+n];

cp's OEIS Frontend

A347396 a(n) = A347395(A276086(n)), where A347395 is Dirichlet convolution of Liouville's lambda with A342001.

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A366803 Dirichlet convolution of Liouville's lambda with A342001 applied to Doudna sequence: a(n) = A347395(A005940(1+n)).

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A342001 Arithmetic derivative without its inherited divisor; the arithmetic derivative of n divided by A003557(n), which is a common divisor of both n and A003415(n).

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A347234 Dirichlet convolution of A126760 with A342001.

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A346485 Möbius transform of A342001, where A342001(n) = A003415(n)/A003557(n).

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

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A369069 Dirichlet convolution of Liouville's lambda (A008836) with A083345, where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.

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A349396 Dirichlet convolution of A342001 ({arithmetic derivative of n}/A003557(n)) with A055615 (Dirichlet inverse of n).

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A366805 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366803(i) = A366803(j) for all i, j >= 0.