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

1,1

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.

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A006530, A007947, A137845.

Sequence in context: A078130 A062171 A048108 * A137845 A046099 A344653

Adjacent sequences: A228954 A228955 A228956 * A228958 A228959 A228960

Keyword

nonn

Author

Michel Lagneau, Sep 09 2013

Status

approved