A072414 Non-Achilles numbers for which LCM of the exponents in the prime factorization of n is not equal to the maximum of the same exponents.

360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1440, 1500, 1656, 1836, 1960, 2016, 2052, 2088, 2160, 2200, 2232, 2250, 2400, 2484, 2520, 2600, 2646, 2664, 2904, 2952, 3024, 3096, 3132, 3168, 3240, 3348, 3384, 3400, 3500

Offset: 1

Author

Labos Elemer, Jun 17 2002

Keywords

• nonn

Comments

Most members of this sequence fail to be Achilles numbers because they have at least one prime factor with multiplicity 1. There are also numbers in the sequence that fail to be Achilles numbers because they are perfect powers: these are precisely the proper powers of members of A072412, so the smallest such is 5184 = 2^6*3^4 = 72^2. - Franklin T. Adams-Watters, Oct 09 2006

Examples

			m = 504 = 2*2*2*3*3*7: exponent-set = E = {3,2,1}, max(E) = 3 < lcm(E) = 6, gcd(E) = min(E) = 1.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005361, A051903, A051904, A052409, A072411, A072412, A052486.

Programs

Mathematica

Select[Range@ 3500, And[LCM @@ # != Max@ #, GCD @@ # == Min@ # == 1] &[FactorInteger[#][[All, -1]] ] &] (* Michael De Vlieger, Jul 18 2017 *)

PARI

is(n)=my(f=factor(n)[,2]); n>9 && lcm(f)!=vecmax(f) && (#f==1 || vecmin(f)<2) \\ Charles R Greathouse IV, Oct 16 2015

Formula

A051903(a(n)) is not equal A072411(a(n)) but the numbers are not in A052486.