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%I #50 May 27 2023 18:24:36
%S 1,3,4,1,6,12,8,1,1,18,12,4,14,24,24,1,18,3,20,6,32,36,24,4,1,42,1,8,
%T 30,72,32,1,48,54,48,1,38,60,56,6,42,96,44,12,6,72,48,4,1,3,72,14,54,
%U 3,72,8,80,90,60,24,62,96,8,1,84,144,68,18,96,144,72,1,74,114,4,20,96,168,80
%N Sum of unitary, squarefree divisors of n, including 1.
%C Unitary convolution of the sequence of n*mu^2(n) (absolute values of A055615) and A000012. - _R. J. Mathar_, May 30 2011
%H Michael De Vlieger, <a href="/A092261/b092261.txt">Table of n, a(n) for n = 1..10000</a>
%H Eckford Cohen, <a href="http://dx.doi.org/10.1007/BF01180473">Arithmetical functions associated with the unitary divisors of an integer</a>, Math. Zeitschr. 74 (1960) 66-80, sequence sigma'(n).
%H Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004. [Cached copy, with permission of the author]
%F Multiplicative with a(p) = p+1 and a(p^e) = 1 for e > 1. - _Vladeta Jovovic_, Feb 22 2004
%F From _Álvar Ibeas_, Mar 06 2015: (Start)
%F a(n) = a(A055231(n)) = A000203(A055231(n)).
%F Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(1-2s)).
%F (End)
%F From _Antti Karttunen_, Nov 25 2017: (Start)
%F a(n) = A048250(A055231(n)).
%F a(n) = A000203(n) / A295294(n).
%F a(n) = A048250(n) / A295295(n) = A048250(n) / A048250(A057521(n)), where A057521(n) = A064549(A003557(n)).
%F (End)
%F Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = Product_{p prime}(1 - 1/(p^2*(p+1))) = 0.881513... (A065465). - _Amiram Eldar_, Jun 10 2020
%F Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(2-3*s) - p^(1-2*s) - p^(2-2*s)). - _Vaclav Kotesovec_, Aug 20 2021
%F a(n) = Sum_{d|n, gcd(d,n/d)=1} d * mu(d)^2. - _Wesley Ivan Hurt_, May 26 2023
%t Table[Plus @@ Select[Divisors@ n, Max @@ Last /@ FactorInteger@ # == 1 && GCD[#, n/#] == 1 &], {n, 1, 79}] (* _Michael De Vlieger_, Mar 08 2015 *)
%t f[p_, e_] := If[e==1, p+1, 1]; a[1]=1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 79] (* _Amiram Eldar_, Mar 01 2019 *)
%o (PARI) a(n) = sumdiv(n, d, d*issquarefree(d)*(gcd(d, n/d) == 1)); \\ _Michel Marcus_, Mar 06 2015
%o (Scheme)
%o ;; This implementation utilizes the memoization-macro definec for which an implementation is available at http://oeis.org/wiki/Memoization#Scheme
%o ;; The other functions, A020639, A067029 and A028234 can be found under the respective entries, and should likewise defined with definec:
%o (definec (A092261 n) (if (= 1 n) 1 (* (+ 1 (if (> (A067029 n) 1) 0 (A020639 n))) (A092261 (A028234 n))))) ;; _Antti Karttunen_, Nov 25 2017
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p^2*X^3 - p*X^2 - p^2*X^2)/(1-X)/(1-p*X))[n], ", ")) \\ _Vaclav Kotesovec_, Aug 20 2021
%Y Cf. A000203, A003557, A007947, A048250, A055231, A056671, A057521, A065465, A295294, A295295, A329728.
%K nonn,mult
%O 1,2
%A _Steven Finch_, Feb 20 2004
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