A347962 -id:A347962 - 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));

A347961 Dirichlet convolution of A342001 with itself.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 4, 1, 2, 0, 14, 0, 2, 2, 10, 0, 14, 0, 18, 2, 2, 0, 42, 1, 2, 4, 22, 0, 40, 0, 20, 2, 2, 2, 63, 0, 2, 2, 58, 0, 48, 0, 30, 20, 2, 0, 92, 1, 18, 2, 34, 0, 40, 2, 74, 2, 2, 0, 204, 0, 2, 24, 35, 2, 64, 0, 42, 2, 56, 0, 162, 0, 2, 20, 46, 2, 72, 0, 132, 10, 2, 0, 260, 2, 2, 2, 106, 0, 210, 2, 54, 2, 2, 2

Offset: 1

Antti Karttunen, Sep 24 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A003415, A003557, A342001, A347962.

Cf. also A305809, A347963.

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));
A347961(n) = sumdiv(n,d,A342001(n/d)*A342001(d));

a(n) = Sum_{d|n} A342001(d) * A342001(n/d).
From Vaclav Kotesovec, Mar 04 2023: (Start)
Let pr(s) = Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s))
and su(s) = Sum_{primes p} p^s/((p^s - 1)*(p^s + p - 1)).
Sum_{k=1..n} a(k) ~ pr(2)^2 * su(2)^2 * Pi^4 * n^2 * log(n) / 72 *
(1 + (2*gamma - 1/2 + 2*pr'(2)/pr(2) + 2*su'(2)/su(2) + 12*zeta'(2)/Pi^2) / log(n)), where
pr(2) = A065464 = 0.428249505677094440218765707581823546121298513355936...
pr'(2) = pr(2) * Sum_{primes p} (3*p - 2) * log(p) / (p^3 - 2*p + 1) = 0.6293283828324697510445630056425352981207558777167836747744750359407...
su(2) = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2)) * PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384...
su'(2) = Sum_{primes p} p^2 * (1-p-p^4) * log(p) / ((p^2-1)^2 * (p^2+p-1)^2) = -0.486606220169261905698805096547122238460686354267440350206456696497...
and gamma is the Euler-Mascheroni constant A001620. (End)

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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