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 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 (list; graph; refs; listen; history; text; internal format)
 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 8^1|16, and 16 with 16^1|16. 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

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Last modified August 5 12:32 EDT 2021. Contains 346467 sequences. (Running on oeis4.)