

A186643


The number of divisors d of n which are either d=1 or for which the highest power d^k dividing n has odd exponent k.


7



1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 5, 2, 4, 4, 3, 2, 5, 2, 5, 4, 4, 2, 8, 2, 4, 4, 5, 2, 8, 2, 5, 4, 4, 4, 6, 2, 4, 4, 8, 2, 8, 2, 5, 5, 4, 2, 8, 2, 5, 4, 5, 2, 8, 4, 8, 4, 4, 2, 11, 2, 4, 5, 5, 4, 8, 2, 5, 4, 8, 2, 10, 2, 4, 5, 5, 4, 8, 2, 8
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OFFSET

1,2


COMMENTS

A divisor d of n is called an "oex divisor" if d=1 or if the highest power d^k dividing n has odd exponent k. a(n) is the number of oex divisors of n.
If q is in A050376, then it is an infinitary divisor of n iff it is an oex divisor of n.
Moreover, every infinitary divisor of n is an oex divisor of n. The converse statement is not generally true.
If d_1 and d_2 are oex divisors of n, then lcm(d_1,d_2) is an oex divisor of n as well.
Not multiplicative: a(2)*a(9) <> a(18), for example.  R. J. Mathar, Mar 25 2012


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000


FORMULA

a(n) >= A037445(n).


EXAMPLE

For n=16, the oex divisors are 1, 8 with 8^116, and 16 with 16^116. Therefore, a(16)=3.


MAPLE

highpp := proc(n, d) if n mod d <> 0 then 0; else nshf := n ; a := 0 ; while nshf mod d = 0 do nshf := nshf /d ; a := a+1 ; end do: a; end if; end proc:
isoex := proc(d, n) d= 1 or (n mod d = 0 and type(highpp(n, d), 'odd') ) ; end proc:
A186643 := proc(n) a := 0 ; for d in numtheory[divisors](n) do if isoex(d, n) then a := a+1 ; end if; end do: a ; end proc: # R. J. Mathar, Mar 18 2011


MATHEMATICA

Table[DivisorSum[n, 1 &, Or[# == 1, OddQ@ IntegerExponent[n, #]] &], {n, 80}] (* Michael De Vlieger, May 28 2017 *)


PROG

(PARI) a(n) = sumdiv(n, d, (d==1)  (valuation(n, d) % 2)); \\ Michel Marcus, Feb 06 2016


CROSSREFS

Cf. A000005, A037445, A050376, A178638.
Sequence in context: A331109 A286324 A318472 * A342087 A286575 A270438
Adjacent sequences: A186640 A186641 A186642 * A186644 A186645 A186646


KEYWORD

nonn,easy


AUTHOR

Vladimir Shevelev, Feb 25 2011


STATUS

approved



