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%I #31 May 28 2017 08:44:55
%S 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,
%T 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,
%U 4,8,2,10,2,4,5,5,4,8,2,8
%N 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.
%C 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.
%C If q is in A050376, then it is an infinitary divisor of n iff it is an oex divisor of n.
%C Moreover, every infinitary divisor of n is an oex divisor of n. The converse statement is not generally true.
%C 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.
%C Not multiplicative: a(2)*a(9) <> a(18), for example. - _R. J. Mathar_, Mar 25 2012
%H Antti Karttunen, <a href="/A186643/b186643.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) >= A037445(n).
%e For n=16, the oex divisors are 1, 8 with 8^1|16, and 16 with 16^1|16. Therefore, a(16)=3.
%p 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:
%p isoex := proc(d,n) d= 1 or (n mod d = 0 and type(highpp(n,d),'odd') ) ; end proc:
%p 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
%t Table[DivisorSum[n, 1 &, Or[# == 1, OddQ@ IntegerExponent[n, #]] &], {n, 80}] (* _Michael De Vlieger_, May 28 2017 *)
%o (PARI) a(n) = sumdiv(n, d, (d==1) || (valuation(n, d) % 2)); \\ _Michel Marcus_, Feb 06 2016
%Y Cf. A000005, A037445, A050376, A178638.
%K nonn,easy
%O 1,2
%A _Vladimir Shevelev_, Feb 25 2011
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