A369454 -id:A369454 - cp's OEIS frontend

A369257 a(n) = number of odd divisors of n that have an even number of prime factors with multiplicity.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 1, 2, 1, 1, 4

Offset: 1

Antti Karttunen, Jan 24 2024

nonn

			Of the eight odd divisors of 105, the four divisors 1, 15, 21, 35 all have an even number of prime factors (A001222(d) is even), therefore a(105) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Inverse Möbius transform of A353557.

Cf. A000265, A001227, A038548, A046337, A053866, A353557, A369258, A369454 (Dirichlet inverse).

A353557(n) = ((n%2)&&(!(bigomega(n)%2)));
A369257(n) = sumdiv(n,d,A353557(d));

a(n) = Sum_{d|n} A353557(d).
a(n) = A001227(n) - A369258(n).
a(n) = a(2*n) = a(A000265(n)).
For n >= 1, a(2n-1) = A038548(2n-1); for n > 1, a(2n) < A038548(2n).
From Antti Karttunen, Jan 27 2024: (Start)
a(n) = A038548(A000265(n)).
a(n) = (A001227(n)+A053866(n))/2.
Dirichlet g.f.: (zeta(s)^2*(1-2^-s) + zeta(2s)*(1+2^-s)) / 2.
(End)

A378526 Dirichlet inverse of A378548, where A378548 is the sum of divisors d of n such that n/d is odd with an even number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, -2, -3, 0, -5, 6, -7, 0, -1, 10, -11, 0, -13, 14, 14, 0, -17, 2, -19, 0, 20, 22, -23, 0, -1, 26, 3, 0, -29, -28, -31, 0, 32, 34, 34, 0, -37, 38, 38, 0, -41, -40, -43, 0, 8, 46, -47, 0, -1, 2, 50, 0, -53, -6, 54, 0, 56, 58, -59, 0, -61, 62, 10, 0, 64, -64, -67, 0, 68, -68, -71, 0, -73, 74, 8, 0, 76, -76, -79, 0, 0, 82

Offset: 1

Antti Karttunen, Dec 01 2024

sign

Agrees with A378525 on all odd n, and also on some even n: 2, 16, 32, 64, 96, 128, 160, 192, ...

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

Cf. A023900, A055615, A353557, A358777, A369454, A378548.

Cf. also A378525, A378527.

A353557(n) = ((n%2)&&(!(bigomega(n)%2)));
A378548(n) = sumdiv(n,d,d*A353557(n/d));
memoA378526 = Map();
A378526(n) = if(1==n,1,my(v); if(mapisdefined(memoA378526,n,&v), v, v = -sumdiv(n,d,if(dA378548(n/d)*A378526(d),0)); mapput(memoA378526,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378548(n/d) * a(d).
a(n) = Sum_{d|n} A023900(d)*A369454(n/d).
a(n) = Sum_{d|n} A055615(d)*A358777(n/d).

A369453 Dirichlet inverse of A038548, where A038548 is the number of divisors of n that are at most sqrt(n).

Original entry on oeis.org

1, -1, -1, -1, -1, 0, -1, 1, -1, 0, -1, 2, -1, 0, 0, 0, -1, 2, -1, 2, 0, 0, -1, 0, -1, 0, 1, 2, -1, 2, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 2, -1, 2, 2, 0, -1, -1, -1, 2, 0, 2, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 2, 0, 0, 2, -1, 2, 0, 2, -1, -3, -1, 0, 2, 2, 0, 2, -1, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0, -1, 2, 2

Offset: 1

Antti Karttunen, Jan 27 2024

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A038548.

Cf. also A359763, A369454.

A038548(n) = if( n<1, 0, sumdiv(n, d, d*d <= n))
memoA369453 = Map();
A369453(n) = if(1==n,1,my(v); if(mapisdefined(memoA369453,n,&v), v, v = -sumdiv(n,d,if(dA038548(n/d)*A369453(d),0)); mapput(memoA369453,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA038548(n/d) * a(d).

Dirichlet g.f.: 2/(zeta(s)^2 + zeta(2*s)).

cp's OEIS Frontend

A369257 a(n) = number of odd divisors of n that have an even number of prime factors with multiplicity.

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A378526 Dirichlet inverse of A378548, where A378548 is the sum of divisors d of n such that n/d is odd with an even number of prime factors (counted with multiplicity).

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A369453 Dirichlet inverse of A038548, where A038548 is the number of divisors of n that are at most sqrt(n).

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