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%I #22 Nov 24 2022 04:52:21
%S 1,1,2,5,4,2,6,7,10,4,10,10,12,6,8,17,16,10,18,20,12,10,22,14,26,12,
%T 26,30,28,8,30,31,20,16,24,50,36,18,24,28,40,12,42,50,40,22,46,34,50,
%U 26,32,60,52,26,40,42,36,28,58,40,60,30,60,65,48,20,66,80
%N The unitary analog of the alternating sum-of-divisors function (A206369).
%C Also equals to the number of integers 1 <= k <= n such that the largest divisor of k which is a unitary divisor of n, gcd(k, n)*, is a square.
%H Amiram Eldar, <a href="/A307037/b307037.txt">Table of n, a(n) for n = 1..10000</a>
%H László Tóth, <a href="https://doi.org/10.2478/ausm-2014-0007">A survey of the alternating sum-of-divisors function</a>, Acta Universitatis Sapientiae, Mathematica, Vol. 5, No. 1 (2013), pp. 93-107, section 6, "Unitary analog".
%F Multiplicative with a(p^e) = p^e + (-1)^e.
%F Equals Sum_{d||n} d*lambda(n/d), where the sum is over the unitary divisors of n, and lambda(n) is Liouville's function (A008836).
%F Sum_{k=1..n} a(k) ~ c*n^2 where c = 63*zeta(3)/(2*Pi^4) = 0.388719...
%F Dirichlet g.f. is zeta(s-1)*zeta(2*s)*zeta(2*s-1) / (zeta(s)*zeta(4*s-2)) (see L. Tóth, A survey ..., p. 101). - _Werner Schulte_, Feb 07 2021
%t f[p_,e_] := p^e + (-1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, e=f[k,2]; f[k,1] = f[k,1]^e+(-1)^e; f[k,2]=1); factorback(f); \\ _Michel Marcus_, Mar 23 2019
%Y Cf. A008836, A034448, A206369.
%K nonn,mult
%O 1,3
%A _Amiram Eldar_, Mar 21 2019
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