A228957 Numbers n such that n/rad(n) is greater than the greatest prime dividing n.

8, 16, 24, 27, 32, 36, 48, 54, 64, 72, 80, 81, 96, 100, 108, 112, 125, 128, 135, 144, 160, 162, 180, 189, 192, 196, 200, 216, 224, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 336, 343, 352, 360, 375, 378, 384, 392, 400, 405, 416, 432, 441, 448, 450, 480

Offset: 1

Author

Michel Lagneau, Sep 09 2013

Keywords

• nonn

Comments

n such that n/rad(n)> gpf(n); numbers n such that n/A007947(n) > A006530(n) where A007947 is the product of the distinct prime factors of n and A006530 is the greatest prime dividing n.

The sequence A137845 (logarithmically smooth numbers)is included in this sequence.

It appears that there exists consecutive numbers such that (80,81), (224,225), (675,676), (1088,1089), (1215,1216), (2375,2376), (2400,2401), (2600, 2601), (3024,3025), (3249,3250), (3968,3969), (4224,4225), (4374,4375), (5831,5832),...

But it appears also that (2400,2401) and (4374,4375) are the only consecutive numbers in the sequence A137845.

Examples

			24 is in the sequence because the prime divisors of 24 are 2 and 3 and 24/2*3 > 3.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006530, A007947, A137845.

A366250 is a subsequence.

Programs

Maple

with(numtheory) :for n from 1 to 400 do:x:=factorset(n):n1:=nops(x): p:= product('x[i]', 'i'=1..n1):m:=n/p:if m> x[n1]then printf(`%d, `,n):else fi:od:

Mathematica

rad[n_]:=Times@@(First@#&/@FactorInteger@n);Select[Range[2,1000],FactorInteger[#][[-1,1]]<#/rad[#]&]
nrQ[n_]:=Module[{x=FactorInteger[n][[All,1]]},n/Times@@x>Last[x]]; Select[Range[ 500],nrQ] (* Harvey P. Dale, Jun 15 2022 *)

PARI

is(n)=my(f=factor(n)); prod(i=1,#f~,f[i,1]^(f[i,2]-1)) > f[#f~,1] \\ Charles R Greathouse IV, Sep 09 2013