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Numbers that are both exponentially odd (A268335) and exponentially odious (A270428).
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%I #9 Dec 01 2023 15:53:57

%S 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,

%T 39,41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,74,77,

%U 78,79,82,83,85,86,87,89,91,93,94,95,97,101,102,103,105,106

%N Numbers that are both exponentially odd (A268335) and exponentially odious (A270428).

%C First differs from its subsequence A005117 at n = 79: a(79) = 128 is not a squarefree number.

%C First differs from A077377 at n = 63, and from A348506 at n = 68.

%C Numbers whose prime factorization contains only exponents that are odd odious numbers (A092246).

%C The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.61156148494581943994..., where f(x) = (1-x) * (1 + x/(2*(1-x^2)) + (Product_{k>=0} (1-(-x)^(2^k)) - Product_{k>=0} (1-x^(2^k))))/2.

%H Amiram Eldar, <a href="/A367801/b367801.txt">Table of n, a(n) for n = 1..12232</a> (terms below 20000)

%H Vladimir Shevelev, <a href="http://dx.doi.org/10.4064/aa8395-5-2016">S-exponential numbers</a>, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.

%t odQ[n_] := OddQ[n] && OddQ[DigitCount[n, 2, 1]]; Select[Range[150], AllTrue[FactorInteger[#][[;;, 2]], odQ] &]

%o (PARI) is(n) = {my(f = factor(n)); for (i = 1, #f~, if(!(f[i, 2]%2 && hammingweight(f[i, 2])%2), return (0))); 1;}

%Y Intersection of A268335 and A270428.

%Y Cf. A367802, A367803, A367804.

%Y Subsequences: A005117, A092759.

%Y Cf. A092246.

%Y Cf. A077377, A348506.

%K nonn,easy,base

%O 1,2

%A _Amiram Eldar_, Dec 01 2023