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A162642
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Number of odd exponents in the canonical prime factorization of n.
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23
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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
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REFERENCES
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R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, to appear.
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LINKS
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Jason Kimberley, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from exponents in factorization of n.
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FORMULA
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a(n) = A001221(n) - A162641(n).
a(n) = A001221(A007913(n)). - Jason Kimberley, Jan 06 2016
a(A000290(n)) = 0, n > 0. - Michel Marcus, Jan 08 2016
G.f.: Sum_{i>=1} Sum_{j>=1} (-1)^j x^(prime(i)^j)/(x^(prime(i)^j) - 1). - Robert Israel, Jan 15 2016
From Antti Karttunen, Nov 28 2017: (Start)
Additive with a(p^e) = A000035(e).
a(n) = A056169(n) + A295662(n).
A056169(n) <= a(n) <= A056169(n) + A295659(n).
a(n) <= A295664(n).
(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])}>;
[A162642(n):n in[1..105]]; // Jason Kimberley, Dec 30 2015
(PARI) a(n) = {my(f = factor(n)); sum(k=1, #f~, f[k, 2] % 2); } \\ Michel Marcus, Jan 08 2016
(Scheme, with memoization-macro definec) (definec (A162642 n) (if (= 1 n) 0 (+ (A000035 (A067029 n)) (A162642 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017
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CROSSREFS
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Cf. A000290 (positions of zeros), A001221, A001620, A002035, A007913, A056169, A162641, A295316, A295659, A295662, A295664.
Sequence in context: A029369 A255315 A125072 * A361205 A355827 A139146
Adjacent sequences: A162639 A162640 A162641 * A162643 A162644 A162645
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KEYWORD
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nonn,easy
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AUTHOR
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Reinhard Zumkeller, Jul 08 2009
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STATUS
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approved
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