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A162642 Number of odd exponents in the canonical prime factorization of n. 16
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

Index entries for sequences computed from exponents in factorization of n

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)

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<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

CROSSREFS

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

Sequence in context: A029369 A255315 A125072 * A139146 A340489 A277487

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 May 7 14:09 EDT 2021. Contains 343650 sequences. (Running on oeis4.)