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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
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
1,2
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
LINKS
Eric Weisstein's World of Mathematics, Divisor Function
Eric Weisstein's World of Mathematics, Greatest Prime Factor
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).
EXAMPLE
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.
MATHEMATICA
Table[Count[Divisors[n], _?(#<=FactorInteger[n][[-1, 1]]&)], {n, 100}] (* Harvey P. Dale, May 01 2016 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Aug 17 2003
STATUS
approved