A087133 Number of divisors of n that are not greater than the greatest prime-factor of n; a(1)=1.

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 2, 4, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 2, 5, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 3, 2, 5, 2, 3, 3, 2, 3, 5, 2, 4, 3, 4, 2, 3, 2, 3, 3, 4, 3, 5, 2, 4, 2, 3, 2, 6, 3, 3, 3, 5, 2, 4, 3, 4, 3, 3, 3, 3, 2, 3, 4, 4

Offset: 1

Author

Reinhard Zumkeller, Aug 17 2003

Keywords

• nonn

Comments

For n > 1, a(n) is the index of the greatest prime factor of n among the divisors of n (see A027750). - Michel Marcus, Jan 21 2019

Examples

			n=28: gpf(28)=7 and divisors = {1,2,4,7,14,28}: 1<=7, 2<=7, 4<=7 and 7<=7, therefore a(28)=4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Eric Weisstein's World of Mathematics, Divisor Function
• Eric Weisstein's World of Mathematics, Greatest Prime Factor

Crossrefs

Cf. A000005, A006530, A027750, A000961, A087134.

Programs

Mathematica

Table[Count[Divisors[n],?(#<=FactorInteger[n][[-1,1]]&)],{n,100}] (* _Harvey P. Dale, May 01 2016 *)

PARI

a(n) = if (n==1, 1, my(f = factor(n), gpf = f[#f~,1]); sumdiv(n, d, d <= gpf)); \\ Michel Marcus, Sep 21 2014

PARI

a(n) = if (n==1, 1, vecsearch(divisors(n), vecmax(factor(n)[,1]))); \\ Michel Marcus, Jan 21 2019

Formula

a(n) <= A000005(n), a(n)=A000005(n) iff n is prime or n=1;

a(n)=2 iff n > 1 is a prime power (A000961);

a(A087134(n))=n and a(k) < n for k < A087134(n).