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A268335 Exponentially odd numbers. 8
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, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence is formed by 1 and the numbers whose prime power factorization contains only odd exponents.

The density of the sequence is the constant given by A065463.

Except for the first term the same as A002035. - R. J. Mathar, Feb 07 2016

Also numbers k all of whose divisors are bi-unitary divisors (i.e., A286324(k) = A000005(k)). - Amiram Eldar, Dec 19 2018

LINKS

Peter J. C. Moses, Table of n, a(n) for n = 1..2000

Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.

Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.

Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.

FORMULA

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).

MATHEMATICA

Select[Range@ 100, AllTrue[Last /@ FactorInteger@ #, OddQ] &] (* Version 10, or *)

Select[Range@ 100, Times @@ Boole[OddQ /@ Last /@ FactorInteger@ #] == 1 &] (* Michael De Vlieger, Feb 02 2016 *)

PROG

(PARI) isok(n)=my(f = factor(n)); for (k=1, #f~, if (!(f[k, 2] % 2), return (0))); 1; \\ Michel Marcus, Feb 02 2016

CROSSREFS

Cf. A002035, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A124010.

Sequence in context: A028801 A239161 A162644 * A002035 A036537 A072510

Adjacent sequences:  A268332 A268333 A268334 * A268336 A268337 A268338

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Feb 01 2016

STATUS

approved

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Last modified July 21 17:43 EDT 2019. Contains 325198 sequences. (Running on oeis4.)