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A372329 a(n) is the smallest multiple of n whose number of divisors is a power of 2 (A036537).

%I A372329 #12 Apr 28 2024 11:35:03
%S A372329 1,2,3,8,5,6,7,8,27,10,11,24,13,14,15,128,17,54,19,40,21,22,23,24,125,
%T A372329 26,27,56,29,30,31,128,33,34,35,216,37,38,39,40,41,42,43,88,135,46,47,
%U A372329 384,343,250,51,104,53,54,55,56,57,58,59,120,61,62,189,128,65
%N A372329 a(n) is the smallest multiple of n whose number of divisors is a power of 2 (A036537).
%H A372329 Amiram Eldar, <a href="/A372329/b372329.txt">Table of n, a(n) for n = 1..10000</a>
%F A372329 Multiplicative with a(p^e) = p^(2^ceiling(log_2(e+1)) - 1).
%F A372329 a(n) = n * A372328(n).
%F A372329 a(n) = n if and only if n is in A036537.
%F A372329 a(n) <= n^2, with equality if and only if n = 1.
%t A372329 f[p_, e_] := p^(2^Ceiling[Log2[e + 1]] - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A372329 (PARI) s(n) = {my(e=logint(n + 1, 2)); if(n + 1 == 2^e, n, 2^(e+1) - 1)};
%o A372329 a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i, 1]^s(f[i, 2]))};
%Y A372329 Cf. A036537, A372328.
%Y A372329 Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193, A356194.
%Y A372329 Differs from A102631 at n = 8, 24, 27, 32, 40, 54, 56, 64, ... .
%K A372329 nonn,easy,mult
%O A372329 1,2
%A A372329 _Amiram Eldar_, Apr 28 2024