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A385043 The sum of the unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

%I A385043 #8 Jun 16 2025 16:54:14
%S A385043 1,3,4,5,6,12,8,1,10,18,12,20,14,24,24,17,18,30,20,30,32,36,24,4,26,
%T A385043 42,1,40,30,72,32,1,48,54,48,50,38,60,56,6,42,96,44,60,60,72,48,68,50,
%U A385043 78,72,70,54,3,72,8,80,90,60,120,62,96,80,1,84,144,68,90,96
%N A385043 The sum of the unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).
%C A385043 The number of these divisors is A385042(n), and the largest of them is A367168(n).
%H A385043 Amiram Eldar, <a href="/A385043/b385043.txt">Table of n, a(n) for n = 1..10000</a>
%F A385043 Multiplicative with a(p^e) = p^(A209229(e)) + 1.
%F A385043 a(n) <= A034448(n), with equality if and only if n is in A138302.
%F A385043 a(n) <= A353900(n), with equality if and only if n is squarefree (A005117).
%F A385043 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1/(p*(p+1)) + Sum_{k>=2} (1/p^(2^k)-1/p^(2^k-1))) = 1.21427559551509410114... .
%t A385043 f[p_, e_] := If[e == 2^IntegerExponent[e, 2], p^e + 1, 1]; a[ 1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385043 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1<<valuation(f[i, 2], 2), f[i, 1]^f[i, 2]+1, 1));}
%Y A385043 The unitary analog of A353900.
%Y A385043 Cf. A005117, A034448, A138302, A209229, A385042.
%Y A385043 The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), this sequence (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).
%K A385043 nonn,easy,mult
%O A385043 1,2
%A A385043 _Amiram Eldar_, Jun 16 2025