A328258 - OEIS

%I #27 Mar 06 2023 10:24:21

%S 1,-1,4,-3,6,-4,8,-7,10,-6,12,-12,14,-8,24,-15,18,-10,20,-18,32,-12,

%T 24,-28,26,-14,28,-24,30,-24,32,-31,48,-18,48,-30,38,-20,56,-42,42,

%U -32,44,-36,60,-24,48,-60,50,-26,72,-42,54,-28,72,-56,80,-30,60,-72,62,-32,80,-63,84

%N a(n) = Sum_{d|n, gcd(d,n/d) = 1} (-1)^(d + 1) * d.

%C Excess of sum of odd unitary divisors of n over sum of even unitary divisors of n.

%C a(n) = n+1 iff n is in A061345 \ {1}. - _Bernard Schott_, Mar 05 2023

%H Robert Israel, <a href="/A328258/b328258.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>.

%F If n = Product (p_j^k_j) then a(n) = Product (1 - (-1)^p_j * p_j^k_j).

%F If n odd, a(n) = usigma(n), where usigma = A034448.

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)/(14*zeta(3)) = A306633 / 14 = 0.0977451... . - _Amiram Eldar_, Nov 17 2022

%F From _Amiram Eldar_, Jan 28 2023: (Start)

%F a(n) = 2 * A192066(n) - A034448(n).

%F a(n) = A192066(n) - A360156(n/2) if n is even, and A192066(n) otherwise.

%F Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1))*(2^(2*s)-2^(s+2)+2)/(2^(2*s)-2). (End)

%p f:= proc(n) local t;

%p   mul(1 - (-1)^t[1] * t[1]^t[2], t=ifactors(n)[2])

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Oct 10 2019

%t a[n_] := Sum[Boole[GCD[d, n/d] == 1] (-1)^(d + 1) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]

%t a[1] = 1; a[n_] := Times @@ (1 - (-1)^First[#] First[#]^Last[#] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]

%o (Magma) [&+[(-1)^(d+1)*d:d in Divisors(n)|Gcd(d, n div d) eq 1]:n in [1..70]]; // _Marius A. Burtea_, Oct 10 2019

%o (PARI) a(n) = sumdiv(n, d, if (gcd(d,n/d) == 1, (-1)^(d + 1) * d)); \\ _Michel Marcus_, Oct 10 2019

%Y Cf. A002129, A034448, A077610, A109712, A192066, A254503, A306633, A360156.

%K sign,mult,look

%O 1,3

%A _Ilya Gutkovskiy_, Oct 09 2019