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A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

%I A385042 #8 Jun 16 2025 16:54:09
%S A385042 1,2,2,2,2,4,2,1,2,4,2,4,2,4,4,2,2,4,2,4,4,4,2,2,2,4,1,4,2,8,2,1,4,4,
%T A385042 4,4,2,4,4,2,2,8,2,4,4,4,2,4,2,4,4,4,2,2,4,2,4,4,2,8,2,4,4,1,4,8,2,4,
%U A385042 4,8,2,2,2,4,4,4,4,8,2,4,2,4,2,8,4,4,4
%N A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).
%C A385042 First differs from A367515 at n = 128.
%C A385042 The sum of these divisors is A385043(n), and the largest of them is A367168(n).
%H A385042 Amiram Eldar, <a href="/A385042/b385042.txt">Table of n, a(n) for n = 1..10000</a>
%F A385042 Multiplicative with a(p^e) = A209229(e) + 1.
%F A385042 a(n) <= A034444(n), with equality if and only if n is in A138302.
%F A385042 a(n) <= A353898(n), with equality if and only if n is squarefree (A005117).
%t A385042 f[p_, e_] := Boole[e == 2^IntegerExponent[e, 2]] + 1; a[ 1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385042 (PARI) a(n) = vecprod(apply(x -> (x == 1<<valuation(x, 2)) + 1, factor(n)[, 2]));
%Y A385042 The unitary analog of A353898.
%Y A385042 Cf. A005117, A034444, A138302, A209229, A367168, A367515, A385043.
%Y A385042 The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), this sequence (exponentially 2^n), A385044 (5-rough).
%K A385042 nonn,easy,mult
%O A385042 1,2
%A A385042 _Amiram Eldar_, Jun 16 2025