A353469 -id:A353469 - cp's OEIS frontend

A353459 Sum of A353457 and its Dirichlet inverse.

2, 0, 0, 1, 0, -2, 0, 1, 1, 2, 0, -1, 0, -2, -2, 1, 0, -1, 0, 1, 2, 2, 0, -1, 1, -2, 1, -1, 0, 0, 0, 1, -2, 2, -2, 0, 0, -2, 2, 1, 0, 0, 0, 1, -1, 2, 0, -1, 1, 1, -2, -1, 0, -1, 2, -1, 2, -2, 0, -1, 0, 2, 1, 1, -2, 0, 0, 1, -2, 0, 0, 0, 0, -2, -1, -1, -2, 0, 0, 1, 1, 2, 0, 1, 2, -2, 2, 1, 0, 1, 2, 1, -2, 2, -2, -1, 0

Offset: 1

Antti Karttunen, Apr 21 2022

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Only values in range {-2, -1, 0, +1, +2} occur.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A348717, A353457, A353458.

Cf. also A353469.

A000265(n) = (n>>valuation(n,2));
A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
memoA353457 = Map();
A353457(n) = if(1==n,1,my(v); if(mapisdefined(memoA353457,n,&v), v, v = -sumdiv(n,d,if(dA353457(A064989(n/d))*A353457(d),0)); mapput(memoA353457,n,v); (v)));
A353459(n) = (A353457(n)+A353457(A064989(n)));

from math import prod
from sympy import factorint, primepi
def A353459(n):
    f = [(primepi(p)&1, -int(e==1)) for p, e in factorint(n).items()]
    return prod(e for p, e in f if not p)+prod(e for p, e in f if p) # Chai Wah Wu, Jan 05 2023

a(n) = A353457(n) + A353458(n) = A353457(n) + A353457(A064989(n)).
For n > 1, a(n) = -Sum_{d|n, 1A353457(d) * A353458(n/d). [As the sequences are Dirichlet inverses of each other]
For all n >= 1, a(n) = a(A003961(n)) = a(A348717(n)).

A353467 a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA252463(n/d)) * a(d).

Original entry on oeis.org

1, -1, 1, 2, -1, -3, 1, -5, -1, 3, -1, 10, 1, -3, 1, 14, -1, 0, 1, -10, -1, 3, -1, -35, 2, -3, 2, 10, 1, 3, -1, -42, 1, 3, -3, 10, 1, -3, -1, 35, -1, -3, 1, -10, -3, 3, -1, 126, -1, -9, 1, 10, 1, -1, 3, -35, -1, -3, -1, -30, 1, 3, 3, 132, -3, 3, -1, -10, 1, 15, 1, -70, -1, -3, -1, 10, 1, -3, 1, -126, -5, 3, -1, 30

Offset: 1

Antti Karttunen, Apr 21 2022

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Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A252463, A353468 [Dirichlet inverse, also a(A252463(n))], A353469 [sum with it].

Cf. also A353457.

A252463(n) = if(!(n%2),n/2,my(f=factor(n)); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f));
memoA353467 = Map();
A353467(n) = if(1==n,1,my(v); if(mapisdefined(memoA353467,n,&v), v, v = -sumdiv(n,d,if(dA353467(A252463(n/d))*A353467(d),0)); mapput(memoA353467,n,v); (v)));

a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA353468(n/d) * a(d).

For all n >= 1, a(A000040(n)) = ((-1)^n).

A353468 Dirichlet inverse of A353467.

Original entry on oeis.org

1, 1, -1, -1, 1, 1, -1, 2, 2, -1, 1, -3, -1, 1, -3, -5, 1, -1, -1, 3, 3, -1, 1, 10, -1, 1, -5, -3, -1, 1, 1, 14, -3, -1, 1, 0, -1, 1, 3, -10, 1, -1, -1, 3, 10, -1, 1, -35, 2, 2, -3, -3, -1, 2, -1, 10, 3, 1, 1, 3, -1, -1, -10, -42, 1, 1, 1, 3, -3, -3, -1, 10, 1, 1, 0, -3, -3, -1, -1, 35, 14, -1, 1, -3, -1, 1, 3, -10

Offset: 1

Antti Karttunen, Apr 21 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A252463, A353467 [Dirichlet inverse], A353469 [sum with it].

Cf. also A353458.

A252463(n) = if(!(n%2),n/2,my(f=factor(n)); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f));
memoA353467 = Map();
A353467(n) = if(1==n,1,my(v); if(mapisdefined(memoA353467,n,&v), v, v = -sumdiv(n,d,if(dA353467(A252463(n/d))*A353467(d),0)); mapput(memoA353467,n,v); (v)));
A353468(n) = A353467(A252463(n));

a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA353467(n/d) * a(d).
a(n) = A353467(A252463(n)).
For all n >= 1, a(A000040(n)) = ((-1)^(n-1)).

cp's OEIS Frontend

A353459 Sum of A353457 and its Dirichlet inverse.

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A353467 a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA252463(n/d)) * a(d).

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A353468 Dirichlet inverse of A353467.

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