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%I #6 Dec 21 2023 21:15:39
%S 1,1,1,4,1,1,1,8,1,1,1,4,1,1,1,16,1,1,1,4,1,1,1,8,1,1,27,4,1,1,1,32,1,
%T 1,1,4,1,1,1,8,1,1,1,4,1,1,1,16,1,1,1,4,1,27,1,8,1,1,1,4,1,1,1,64,1,1,
%U 1,4,1,1,1,8,1,1,1,4,1,1,1,16,81,1,1,4,1
%N The largest term of A054744 that divide n.
%C The largest divisor d of n such that e >= p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).
%H Amiram Eldar, <a href="/A368333/b368333.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = p^e if e >= p.
%F A034444(a(n)) = A368334(n).
%F a(n) >= 1, with equality if and only if n is in A048103.
%F a(n) <= n, with equality if and only if n is in A054744.
%F Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^(p*s) + 1/p^(p*(s-1)) + 1/p^((p+1)*s-1) - 1/p^((p+1)*(s-1)+1)).
%t f[p_, e_] := If[e < p, 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i,1]^f[i,2]));}
%Y Cf. A034444, A048103, A054744, A327939, A368329, A368332, A368334, A368335.
%K nonn,easy,mult
%O 1,4
%A _Amiram Eldar_, Dec 21 2023
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