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A162642
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Number of odd exponents in the canonical prime factorization of n.
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32
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0, 1, 1, 0, 1, 2, 1, 1, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 2, 0, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 0, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 0, 1, 3, 1, 2, 3
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OFFSET
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1,6
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COMMENTS
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a(n) is also known as the squarefree rank of n. - Jason Kimberley, Jul 08 2017
The number of primes that are infinitary divisors of n. - Amiram Eldar, Oct 01 2023
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LINKS
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R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, Journal of Combinatorial Mathematics and Combinatorial Computing, Vol. 106 (2018).
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FORMULA
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G.f.: Sum_{i>=1} Sum_{j>=1} (-1)^j x^(prime(i)^j)/(x^(prime(i)^j) - 1). - Robert Israel, Jan 15 2016
(End)
Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = gamma + Sum_{p prime} (log(1-1/p) + 1/(p+1)) = A077761 - A179119 = -0.0687327134... and gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021
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MAPLE
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A162642 := proc(n) add ( op(2, f) mod 2 , f=ifactors(n)[2]) ; end proc: # R. J. Mathar, Mar 30 2011
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MATHEMATICA
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{0}~Join~Table[Count[Last /@ FactorInteger@ n, e_ /; OddQ@ e], {n, 2, 105}] (* Michael De Vlieger, Jan 06 2016 *)
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PROG
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(Magma) A162642:=func<n|#{pe:pe in Factorisation(n)|IsOdd(pe[2])}>;
(PARI) a(n) = {my(f = factor(n)); sum(k=1, #f~, f[k, 2] % 2); } \\ Michel Marcus, Jan 08 2016
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CROSSREFS
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Cf. A000290 (positions of zeros), A001221, A001620, A002035, A007913, A056169, A162641, A295316, A295659, A295662, A295664.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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