

A319997


a(n) = Sum_{dn, d is odd} mu(n/d)*d, where mu(n) is Moebius function A008683.


5



1, 1, 2, 0, 4, 2, 6, 0, 6, 4, 10, 0, 12, 6, 8, 0, 16, 6, 18, 0, 12, 10, 22, 0, 20, 12, 18, 0, 28, 8, 30, 0, 20, 16, 24, 0, 36, 18, 24, 0, 40, 12, 42, 0, 24, 22, 46, 0, 42, 20, 32, 0, 52, 18, 40, 0, 36, 28, 58, 0, 60, 30, 36, 0, 48, 20, 66, 0, 44, 24, 70, 0, 72, 36, 40, 0, 60, 24, 78, 0, 54, 40, 82, 0, 64, 42, 56, 0, 88
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OFFSET

1,3


LINKS

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


FORMULA

a(n) = Sum_{dn} A000035(d)*A008683(n/d)*d.
a(n) = A000010(n)  A319998(n).
For even n, a(n) = A000010(n)  2*A000010(n/2); for odd n, a(n) = A000010(n).
a(2n+1) = A000010(2n+1), a(4n+2) = A000010(4n+2), a(4n) = 0.
Multiplicative with a(2^1) = 1, a(2^e) = 0 for e > 1, and a(p^e) = (p  1)*p^(e1) when p is an odd prime.
G.f.: Sum_{k>=1} mu(k)*x^k*(1 + x^(2*k))/(1  x^(2*k))^2.  Ilya Gutkovskiy, Nov 02 2018


PROG

(PARI) A319997(n) = sumdiv(n, d, (d%2)*moebius(n/d)*d);
(PARI) A319997(n) = if(n%2, eulerphi(n), if(n%4, eulerphi(n), 0));
(PARI) A319997(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i, 1], (1==f[i, 2]), (f[i, 1]1)*(f[i, 1]^(f[i, 2]1)))); };


CROSSREFS

Cf. A000010, A062570, A319998.
Sequence in context: A328599 A222303 A097945 * A153733 A083218 A203908
Adjacent sequences: A319994 A319995 A319996 * A319998 A319999 A320000


KEYWORD

sign,mult


AUTHOR

Antti Karttunen, Oct 31 2018


STATUS

approved



