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A270428 Exponentially odious numbers: 1 together with positive integers n such that all exponents in prime factorization of n are odious numbers (A000069).

%I A270428 #32 Mar 08 2025 18:38:17
%S A270428 1,2,3,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,28,
%T A270428 29,30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U A270428 55,57,58,59,60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,81,82,83,84
%N A270428 Exponentially odious numbers: 1 together with positive integers n such that all exponents in prime factorization of n are odious numbers (A000069).
%C A270428 A268385 maps each term of this sequence to a unique term of A268335, and vice versa.
%C A270428 The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.87686263163054480657..., where f(x) = 1 - x + (1 - (1-x) * Product_{k>=0} (1-x^(2^k)))/2. - _Amiram Eldar_, Oct 27 2023
%H A270428 Antti Karttunen, <a href="/A270428/b270428.txt">Table of n, a(n) for n = 1..10000</a>
%H A270428 Vladimir Shevelev, <a href="http://dx.doi.org/10.4064/aa8395-5-2016">S-exponential numbers</a>, Acta Arithmetica, Vol. 175 (2016), 385-395.
%t A270428 odiousQ[n_] := OddQ[DigitCount[n, 2, 1]]; Select[Range[100], AllTrue[FactorInteger[#][[;;, 2]], odiousQ] &] (* _Amiram Eldar_, May 18 2023 *)
%o A270428 (PARI)
%o A270428 A355825(n) = factorback(apply(e->(hammingweight(e)%2), factor(n)[, 2]));
%o A270428 isA270428(n) = A355825(n); \\ _Antti Karttunen_, Jul 21 2022
%o A270428 (Scheme)
%o A270428 ;; Requires _Antti Karttunen_'s IntSeq-library.
%o A270428 (define A270428 (ZERO-POS 1 1 (COMPOSE sub1 A270419)))
%o A270428 (Scheme)
%o A270428 ;; Requires _Antti Karttunen_'s IntSeq-library.
%o A270428 (definec (chA270428 n) (cond ((= 1 n) 1) (else (* (A010060 (A067029 n)) (chA270428 (A028234 n))))))
%o A270428 (define A270428 (NONZERO-POS 1 1 chA270428))
%Y A270428 Apart from 1, a subsequence of A270420.
%Y A270428 Indices of ones in A270419.
%Y A270428 Sequence A270436 sorted into ascending order.
%Y A270428 Cf. A010060, A028234, A067029, A355825 (characteristic function).
%Y A270428 Cf. also A262675, A268335, A268385.
%Y A270428 Differs from its subsequence A138302 for the first time at n=113, where a(113) = 128 = 2^7, a value which does not occur in A138302.
%K A270428 nonn,easy
%O A270428 1,2
%A A270428 _Antti Karttunen_, May 26 2016