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A087133 Number of divisors of n that are not greater than the greatest prime-factor of n; a(1)=1. 2

%I #20 Jan 22 2019 05:11:30

%S 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,

%T 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,

%U 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

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

%C 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

%H Antti Karttunen, <a href="/A087133/b087133.txt">Table of n, a(n) for n = 1..65537</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>

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

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

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

%e 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.

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

%o (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

%o (PARI) a(n) = if (n==1, 1, vecsearch(divisors(n), vecmax(factor(n)[,1]))); \\ _Michel Marcus_, Jan 21 2019

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

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Aug 17 2003

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