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%I #62 Jan 22 2024 01:23:43
%S 1,2,3,5,6,7,8,10,11,13,14,15,17,19,21,22,23,24,26,27,29,30,31,32,33,
%T 34,35,37,38,39,40,41,42,43,46,47,51,53,54,55,56,57,58,59,61,62,65,66,
%U 67,69,70,71,73,74,77,78,79,82,83,85,86,87,88,89,91,93,94,95,96,97
%N Exponentially odd numbers.
%C The sequence is formed by 1 and the numbers whose prime power factorization contains only odd exponents.
%C The density of the sequence is the constant given by A065463.
%C Except for the first term the same as A002035. - _R. J. Mathar_, Feb 07 2016
%C Also numbers k all of whose divisors are bi-unitary divisors (i.e., A286324(k) = A000005(k)). - _Amiram Eldar_, Dec 19 2018
%C The term "exponentially odd integers" was apparently coined by Cohen (1960). These numbers were also called "unitarily 2-free", or "2-skew", by Cohen (1961). - _Amiram Eldar_, Jan 22 2024
%H Peter J. C. Moses, <a href="/A268335/b268335.txt">Table of n, a(n) for n = 1..2000</a>
%H Eckford Cohen, <a href="https://eudml.org/doc/169870">Arithmetical functions associated with the unitary divisors of an integer</a>, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.
%H Eckford Cohen, <a href="http://pub.acta.hu/acta/showCustomerArticle.action?id=7038&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=3fe5dd60a67f39f9">Some sets of integers related to the k-free integers</a>, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1510.05914">Exponentially S-numbers</a>, arXiv:1510.05914 [math.NT], 2015.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1511.03860">Set of all densities of exponentially S-numbers</a>, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
%H Vladimir Shevelev, <a href="http://dx.doi.org/10.4064/aa8395-5-2016">S-exponential numbers</a>, Acta Arithmetica, Vol. 175(2016), 385-395.
%H D. Suryanarayana and R. Sita Rama Chandra Rao, <a href="https://doi.org/10.1017/S1446788700020425">Distribution of unitarily k-free integers</a>, Journal of the Australian Mathematical Society, Vol. 20 , No. 2 (1975), pp. 129-141.
%F Sum_{a(n)<=x} 1 = C*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c = 4*sqrt(2.4/log 2) = 7.44308... and C = Product_{prime p} (1 - 1/p*(p + 1)) = 0.7044422009991... (A065463).
%F Sum_{n>=1} 1/a(n)^s = zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)), s>1. - _Amiram Eldar_, Sep 26 2023
%t Select[Range@ 100, AllTrue[Last /@ FactorInteger@ #, OddQ] &] (* Version 10, or *)
%t Select[Range@ 100, Times @@ Boole[OddQ /@ Last /@ FactorInteger@ #] == 1 &] (* _Michael De Vlieger_, Feb 02 2016 *)
%o (PARI) isok(n)=my(f = factor(n)); for (k=1, #f~, if (!(f[k,2] % 2), return (0))); 1; \\ _Michel Marcus_, Feb 02 2016
%o (Python)
%o from itertools import count, islice
%o from sympy import factorint
%o def A268335_gen(startvalue=1): # generator of terms >= startvalue
%o return filter(lambda n:all(e&1 for e in factorint(n).values()),count(max(startvalue,1)))
%o A268335_list = list(islice(A268335_gen(),20)) # _Chai Wah Wu_, Jun 22 2023
%Y Cf. A002035, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A124010.
%K nonn
%O 1,2
%A _Vladimir Shevelev_, Feb 01 2016