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A366903 The sum of exponentially odious divisors of n.

%I A366903 #7 Oct 28 2023 03:46:48
%S A366903 1,3,4,7,6,12,8,7,13,18,12,28,14,24,24,23,18,39,20,42,32,36,24,28,31,
%T A366903 42,13,56,30,72,32,23,48,54,48,91,38,60,56,42,42,96,44,84,78,72,48,92,
%U A366903 57,93,72,98,54,39,72,56,80,90,60,168,62,96,104,23,84,144,68
%N A366903 The sum of exponentially odious divisors of n.
%C A366903 First differs from A353900 at n = 128.
%C A366903 The number of these divisors is A366901(n) and the largest of them is A366905(n).
%H A366903 Amiram Eldar, <a href="/A366903/b366903.txt">Table of n, a(n) for n = 1..10000</a>
%F A366903 Multiplicative with a(p^e) = 1 + Sum_{k = 1..e, k is odious} p^k.
%F A366903 a(n) <= A000203(n), with equality if and only if n is a cubefree number (A004709).
%F A366903 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1-1/p)*(1 + Sum_{k>=1} a(p^k)/p^(2*k)) = 0.721190607... .
%t A366903 f[p_, e_] := 1 + Total[p^Select[Range[e], OddQ[DigitCount[#, 2, 1]] &]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366903 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sum(k = 1, f[i, 2], (hammingweight(k)%2) * f[i, 1]^k));}
%Y A366903 Cf. A004709, A000203, A366901, A366905.
%Y A366903 Similar sequences: A353900, A365682, A366904.
%K A366903 nonn,easy,mult
%O A366903 1,2
%A A366903 _Amiram Eldar_, Oct 27 2023