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A350388 a(n) is the largest unitary divisor of n that is a square.

%I A350388 #22 Jun 01 2025 03:24:34
%S A350388 1,1,1,4,1,1,1,1,9,1,1,4,1,1,1,16,1,9,1,4,1,1,1,1,25,1,1,4,1,1,1,1,1,
%T A350388 1,1,36,1,1,1,1,1,1,1,4,9,1,1,16,49,25,1,4,1,1,1,1,1,1,1,4,1,1,9,64,1,
%U A350388 1,1,4,1,1,1,9,1,1,25,4,1,1,1,16,81,1,1,4
%N A350388 a(n) is the largest unitary divisor of n that is a square.
%C A350388 First differs from A056623 at n = 32.
%H A350388 Amiram Eldar, <a href="/A350388/b350388.txt">Table of n, a(n) for n = 1..10000</a>
%F A350388 Multiplicative with a(p^e) = p^e if e is even and 1 otherwise.
%F A350388 a(n) = n/A350389(n).
%F A350388 a(n) = A071974(n)^2.
%F A350388 a(n) = A008833(n) if and only if n is in A335275.
%F A350388 A001222(a(n)) = A350386(n).
%F A350388 a(n) = 1 if and only if n is an exponentially odd number (A268335).
%F A350388 a(n) = n if and only if n is a positive square (A000290 \ {0}).
%F A350388 Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (1/3) * Product_{p prime} (1 + sqrt(p)/(1 + p + p^2)) = 0.59317173657411718128... [updated Oct 16 2022]
%F A350388 Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) - 1/p^(3*s-2)). - _Amiram Eldar_, Oct 01 2023
%F A350388 Sum_{d|n, gcd(d, n/d) == 1} A076479(d) * a(n/d) = A191414(sqrt(n)) if n is a square, and 0 otherwise. - _Amiram Eldar_, Jun 01 2025
%t A350388 f[p_, e_] := If[EvenQ[e], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A350388 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, 1, f[i,1]^f[i,2]));} \\ _Amiram Eldar_, Oct 01 2023
%Y A350388 Cf. A000290, A001222, A008833, A056623, A071974, A076479, A191414, A268335, A335275, A350386, A350389.
%K A350388 nonn,easy,mult
%O A350388 1,4
%A A350388 _Amiram Eldar_, Dec 28 2021