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A360969 Multiplicative with a(p^e) = e^2, p prime and e > 0.

%I A360969 #26 Mar 30 2023 02:39:14
%S A360969 1,1,1,4,1,1,1,9,4,1,1,4,1,1,1,16,1,4,1,4,1,1,1,9,4,1,9,4,1,1,1,25,1,
%T A360969 1,1,16,1,1,1,9,1,1,1,4,4,1,1,16,4,4,1,4,1,9,1,9,1,1,1,4,1,1,4,36,1,1,
%U A360969 1,4,1,1,1,36,1,1,4,4,1,1,1,16,16,1,1,4
%N A360969 Multiplicative with a(p^e) = e^2, p prime and e > 0.
%C A360969 From _Bernard Schott_, Feb 27 2023: (Start)
%C A360969 The three fixed points are 1, 4 and 16.
%C A360969 a(n) = 1 iff n is A005117.
%C A360969 a(n) = 4 iff n is in A060687. (End)
%H A360969 Robert Israel, <a href="/A360969/b360969.txt">Table of n, a(n) for n = 1..10000</a>
%F A360969 Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + (3*p^s - 1) / (p^s*(p^s - 1)^2)).
%F A360969 Sum_{k=1..n} a(k) ~ c*n, where c = Product_{primes p} (1 + (3*p - 1) / (p*(p-1)^2)) = 8.18840474382698544967326709964388539461401085196013492328186138...
%F A360969 a(n) = A005361(n)^2.
%p A360969 f:= proc(n) local t;
%p A360969   mul(t^2, t = ifactors(n)[2][..,2]);
%p A360969 end proc:
%p A360969 map(f, [$1..100]); # _Robert Israel_, Mar 29 2023
%t A360969 g[p_, e_] := e^2; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
%o A360969 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + 4*X^2 - X^3)/(1-X)^3)[n], ", "))
%o A360969 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 1]=f[k,2]^2; f[k, 2]=1); factorback(f); \\ _Michel Marcus_, Feb 27 2023
%Y A360969 Cf. A005361, A322327, A360970, A361132.
%Y A360969 Cf. A005117, A060687.
%K A360969 nonn,mult
%O A360969 1,4
%A A360969 _Vaclav Kotesovec_, Feb 27 2023