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A319997 a(n) = Sum_{d|n, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

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

FORMULA

a(n) = Sum_{d|n} 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^(e-1) 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

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Last modified December 12 20:12 EST 2019. Contains 329961 sequences. (Running on oeis4.)