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.

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

Offset: 1

Author

Vladimir Shevelev, Feb 25 2011

Keywords

• nonn
• easy

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

Examples

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

Links

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

Crossrefs

Cf. A000005, A037445, A050376, A178638.

Programs

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 *)

PARI

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

Formula

a(n) >= A037445(n).