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 A162642 Number of odd exponents in the canonical prime factorization of n. 23
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS a(n) is also known as the squarefree rank of n. - Jason Kimberley, Jul 08 2017 REFERENCES R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, to appear. LINKS Jason Kimberley, Table of n, a(n) for n = 1..20000 FORMULA 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 MAPLE A162642 := proc(n) add ( op(2, f) mod 2 , f=ifactors(n)[2]) ; end proc: # R. J. Mathar, Mar 30 2011 MATHEMATICA {0}~Join~Table[Count[Last /@ FactorInteger@ n, e_ /; OddQ@ e], {n, 2, 105}] (* Michael De Vlieger, Jan 06 2016 *) PROG (Magma) A162642:=func; [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 CROSSREFS 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 KEYWORD nonn,easy AUTHOR Reinhard Zumkeller, Jul 08 2009 STATUS approved

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Last modified March 29 15:03 EDT 2023. Contains 361599 sequences. (Running on oeis4.)