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A162642 Number of odd exponents in the canonical prime factorization of n. 13

%I

%S 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,

%T 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,

%U 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

%N Number of odd exponents in the canonical prime factorization of n.

%C a(n) is also known as the squarefree rank of n. - _Jason Kimberley_, Jul 08 2017

%D R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, to appear.

%H Jason Kimberley, <a href="/A162642/b162642.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = A001221(n) - A162641(n).

%F a(n) = A001221(A007913(n)). - _Jason Kimberley_, Jan 06 2016

%F a(A000290(n)) = 0, n > 0. - _Michel Marcus_, Jan 08 2016

%F G.f.: Sum_{i>=1} Sum_{j>=1} (-1)^j x^(prime(i)^j)/(x^(prime(i)^j) - 1). - _Robert Israel_, Jan 15 2016

%F From _Antti Karttunen_, Nov 28 2017: (Start)

%F Additive with a(p^e) = A000035(e).

%F a(n) = A056169(n) + A295662(n).

%F A056169(n) <= a(n) <= A056169(n) + A295659(n).

%F a(n) <= A295664(n).

%F (End)

%p A162642 := proc(n) add ( op(2,f) mod 2 ,f=ifactors(n)[2]) ; end proc: # _R. J. Mathar_, Mar 30 2011

%t {0}~Join~Table[Count[Last /@ FactorInteger@ n, e_ /; OddQ@ e], {n, 2, 105}] (* _Michael De Vlieger_, Jan 06 2016 *)

%o (MAGMA) A162642:=func<n|#{pe:pe in Factorisation(n)|IsOdd(pe[2])}>;

%o [A162642(n):n in[1..105]]; // _Jason Kimberley_, Dec 30 2015

%o (PARI) a(n) = {my(f = factor(n)); sum(k=1, #f~, f[k,2] % 2);} \\ _Michel Marcus_, Jan 08 2016

%o (Scheme, with memoization-macro definec) (definec (A162642 n) (if (= 1 n) 0 (+ (A000035 (A067029 n)) (A162642 (A028234 n))))) ;; _Antti Karttunen_, Nov 28 2017

%Y Cf. A000290 (positions of zeros), A001221, A002035, A007913, A056169, A162641, A295316, A295659, A295662, A295664.

%K nonn,easy

%O 1,6

%A _Reinhard Zumkeller_, Jul 08 2009

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Last modified September 15 22:10 EDT 2019. Contains 327088 sequences. (Running on oeis4.)