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%I #13 Nov 24 2022 03:16:00
%S 1,3,4,5,6,12,8,15,10,18,12,20,14,24,24,17,18,30,20,30,32,36,24,60,26,
%T 42,40,40,30,72,32,33,48,54,48,50,38,60,56,90,42,96,44,60,60,72,48,68,
%U 50,78,72,70,54,120,72,120,80,90,60,120,62,96,80,85,84,144
%N The sum of the tri-unitary divisors of n.
%C A divisor d of n is tri-unitary if the greatest common bi-unitary divisor of d and n/d is 1.
%H Antti Karttunen, <a href="/A324706/b324706.txt">Table of n, a(n) for n = 1..20000</a>
%H Graeme L. Cohen, <a href="https://doi.org/10.1090/S0025-5718-1990-0993927-5">On an integer's infinitary divisors</a>, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
%H Pentti Haukkanen, <a href="https://www.fq.math.ca/Scanned/38-5/haukkanen.pdf">On the k-ary convolution of arithmetical functions</a>, The Fibonacci Quarterly, Vol. 38, No. 5 (2000) pp. 440-445.
%F Multiplicative with a(p^3) = 1 + p + p^2 + p^3, a(p^6) = 1 + p^2 + p^4 + p^6, and a(p^e) = 1 + p^e otherwise.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^3 + 1/p^4 - 2/p^6 + 2/p^8 - 1/p^9 - 1/p^12 + 1/p^13) = 0.72189237802... . - _Amiram Eldar_, Nov 24 2022
%t f[p_, e_] := If[e == 3, (p^4-1)/(p-1), If[e==6, (p^8-1)/(p^2-1), p^e+1]]; a[1]=1; a[n_]:= Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) A324706(n) = { my(f = factor(n)); prod(i=1, #f~, if(3==f[i,2], sigma(f[i,1]^f[i,2]), if(6==f[i,2], ((f[i,1]^8)-1)/((f[i,1]^2)-1), 1+(f[i,1]^f[i,2])))); }; \\ _Antti Karttunen_, Mar 12 2019
%Y Cf. A000203, A034448, A049417, A072691, A188999, A322485, A324707, A324708, A324709.
%K nonn,mult
%O 1,2
%A _Amiram Eldar_, Mar 11 2019
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