cp's OEIS Frontend

A372328 a(n) is the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).

%I A372328 #9 Apr 28 2024 11:34:47
%S A372328 1,1,1,2,1,1,1,1,3,1,1,2,1,1,1,8,1,3,1,2,1,1,1,1,5,1,1,2,1,1,1,4,1,1,
%T A372328 1,6,1,1,1,1,1,1,1,2,3,1,1,8,7,5,1,2,1,1,1,1,1,1,1,2,1,1,3,2,1,1,1,2,
%U A372328 1,1,1,3,1,1,5,2,1,1,1,8,27,1,1,2,1,1,1
%N A372328 a(n) is the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).
%C A372328 First differs from A331738 at n = 32.
%C A372328 The largest divisor d of n that is infinitarily relatively prime to n (see A064379), i.e., d have no common infinitary divisors with n.
%H A372328 Amiram Eldar, <a href="/A372328/b372328.txt">Table of n, a(n) for n = 1..10000</a>
%F A372328 Multiplicative with a(p^e) = p^(2^ceiling(log_2(e+1)) - e - 1).
%F A372328 a(n) = A372329(n)/n.
%F A372328 a(n) = 1 if and only if n is in A036537.
%F A372328 a(n) <= n, with equality if and only if n = 1.
%t A372328 f[p_, e_] := p^(2^Ceiling[Log2[e + 1]] - e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A372328 (PARI) s(n) = {my(e = logint(n + 1, 2)); if(n + 1 == 2^e, 0, 2^(e+1) - n - 1)};
%o A372328 a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]))};
%Y A372328 Cf. A036537, A064379, A077609, A331738, A372329.
%Y A372328 Similar sequences: A365297, A365298, A365637, A365685, A367931, A367932.
%K A372328 nonn,easy,mult
%O A372328 1,4
%A A372328 _Amiram Eldar_, Apr 28 2024