A349441 -id:A349441 - cp's OEIS frontend

A349443 a(n) = A349441(n) + A349442(n).

2, 0, 0, 1, 0, 4, 0, -3, 4, 8, 0, -6, 0, 12, 16, -1, 0, -8, 0, -12, 24, 20, 0, -6, 16, 24, -16, -18, 0, 0, 0, 5, 40, 32, 48, 14, 0, 36, 48, -12, 0, 0, 0, -30, -32, 44, 0, -2, 36, -24, 64, -36, 0, -16, 80, -18, 72, 56, 0, 8, 0, 60, -48, 7, 96, 0, 0, -48, 88, 0, 0, 6, 0, 72, -48, -54, 120, 0, 0, -4, -20, 80, 0, 12

Offset: 1

Antti Karttunen, Nov 18 2021

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

Cf. A349441, A349442.

A349443(n) = (A349441(n)+A349442(n)); \\ Needs also code from A349441 and A349442.

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349441(d) * A349442(n/d). [As the sequences are Dirichlet inverses of each other]

A349442 Dirichlet convolution of A000027 (the identity function) with A349350 (Dirichlet inverse of the powerful part of n).

Original entry on oeis.org

1, 1, 2, -1, 4, 2, 6, -3, -2, 4, 10, -2, 12, 6, 8, -1, 16, -2, 18, -4, 12, 10, 22, -6, -4, 12, -16, -6, 28, 8, 30, 5, 20, 16, 24, 2, 36, 18, 24, -12, 40, 12, 42, -10, -8, 22, 46, -2, -6, -4, 32, -12, 52, -16, 40, -18, 36, 28, 58, -8, 60, 30, -12, 7, 48, 20, 66, -16, 44, 24, 70, 6, 72, 36, -8, -18, 60, 24, 78, -4

Offset: 1

Antti Karttunen, Nov 18 2021

Multiplicative because both A000027 and A349350 are.

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

Cf. A000027, A057521, A349350, A349441 (Dirichlet inverse), A349443 (sum with it).

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
memoA349350 = Map();
A349350(n) = if(1==n,1,my(v); if(mapisdefined(memoA349350,n,&v), v, v = -sumdiv(n,d,if(dA057521(n/d)*A349350(d),0)); mapput(memoA349350,n,v); (v)));
A349442(n) = sumdiv(n,d,d*A349350(n/d));

a(n) = Sum_{d|n} d * A349350(n/d).

A349379 Möbius transform of A057521 (powerful part of n).

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 18, 0, 0, 0, 0, 16, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0

Offset: 1

Antti Karttunen, Nov 18 2021

Multiplicative with a(p^e) = 0 if e = 1, p^2 - 1 if e = 2 and p^e - p^(e-1) otherwise. - Amiram Eldar, Nov 18 2021

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

Cf. A000010, A008683, A057521, A349441.

Cf. also A300717.

f[p_, e_] := Which[e > 2, p^e - p^(e - 1), e == 2, p^2 - 1, e == 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A349379(n) = sumdiv(n,d,moebius(n/d)*A057521(d));

from math import prod
from sympy import factorint
def A349379(n): return prod(0 if e==1 else p**e - (1 if e==2 else p**(e-1)) for p,e in factorint(n).items()) # Chai Wah Wu, Nov 14 2022

a(n) = Sum_{d|n} A008683(n/d) * A057521(d).

a(n) = Sum_{d|n} A000010(n/d) * A349441(d).

cp's OEIS Frontend

A349443 a(n) = A349441(n) + A349442(n).

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A349442 Dirichlet convolution of A000027 (the identity function) with A349350 (Dirichlet inverse of the powerful part of n).

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A349379 Möbius transform of A057521 (powerful part of n).

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