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The smallest unitary divisor d of n such that n/d is a term of A138302.
1

%I #15 Dec 30 2023 16:04:16

%S 1,1,1,1,1,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,8,1,1,27,1,1,1,1,32,1,

%T 1,1,1,1,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,27,1,8,1,1,1,1,1,1,1,64,1,1,

%U 1,1,1,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,8,1,1,1,1,1,1,1,32,1,1,1,1,1,1,1,8,1,1,1,27,1,1,1,1,1,1,1,1,1,1,1,8,1,1,1,1,125,1,1

%N The smallest unitary divisor d of n such that n/d is a term of A138302.

%C First differs from A368167 at n = 64 and from A367513 at n = 128.

%H Amiram Eldar, <a href="/A368540/b368540.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n / A367168(n).

%F Multiplicative with a(p^e) = p^(e-A048298(e)).

%F a(n) >= 1, with equality if and only if n is in A138302.

%t f[p_, e_] := If[e == 2^IntegerExponent[e, 2], 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] == 1 << valuation(f[i, 2], 2), 1, f[i, 1]^f[i, 2]));}

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A368540(n): return prod(p**e for p, e in factorint(n).items() if not e or (e&-e)^e) # _Chai Wah Wu_, Dec 30 2023

%Y Cf. A048298, A077610, A138302, A367168, A367513, A368167.

%K nonn,easy,mult

%O 1,8

%A _Amiram Eldar_, Dec 29 2023