A062770 - OEIS

%I #29 Aug 19 2024 13:16:52

%S 2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,30,31,

%T 32,33,34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,

%U 65,66,67,69,70,71,73,74,77,78,79,81,82,83,85,86,87,89,91,93,94,95,97,100

%N n/[largest power of squarefree kernel] equals 1; perfect powers of sqf-kernels (or sqf-numbers).

%C The sequence contains numbers m such that the exponents e are identical for all prime power factors p^e | m. It is clear from this alternate definition that m / K^E = 1 iff E is an integer. - _Michael De Vlieger_, Jun 24 2022

%H Michael De Vlieger, <a href="/A062770/b062770.txt">Table of n, a(n) for n = 1..10000</a>

%F A062760(a(n)) = 1, i.e., a(n)/(A007947(a(n))^A051904(a(n))) = a(n)/A062759(a(n)) = 1.

%F a(n) = A072774(n+1). - _Chai Wah Wu_, Aug 19 2024

%e Primes, squarefree numbers and perfect powers are here.

%e From _Michael De Vlieger_, Jun 24 2022 (Start):

%e 144 cannot be in the sequence, since the exponents of its prime power factors differ. The squarefree kernel of 144 = 2^4 * 3^2 is 2*3 = 6. The largest power of 6 less than 144 is 36. 144/36 = 4, so it is not in the sequence.

%e 216 is in the sequence because 216 = 2^3 * 3^3 is 2*3 = 6. But 216 = 6^3, hence 6^3 / 6^3 = 1. (End)

%t Select[Range[2, 2^16], Length@ Union@ FactorInteger[#][[All, -1]] == 1 &] _Michael De Vlieger_, Jun 24 2022

%o (PARI) is(n)=ispower(n,,&n); issquarefree(n) && n>1 \\ _Charles R Greathouse IV_, Sep 18 2015

%o (PARI) is(n)=#Set(factor(n)[,2])==1 \\ _Charles R Greathouse IV_, Sep 18 2015

%o (Python)

%o from math import isqrt

%o from sympy import mobius, integer_nthroot

%o def A062770(n):

%o     def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))

%o     def f(x): return n-2+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(1,y))

%o     kmin, kmax = 1,2

%o     while f(kmax) >= kmax:

%o         kmax <<= 1

%o     while True:

%o         kmid = kmax+kmin>>1

%o         if f(kmid) < kmid:

%o             kmax = kmid

%o         else:

%o             kmin = kmid

%o         if kmax-kmin <= 1:

%o             break

%o     return kmax # _Chai Wah Wu_, Aug 19 2024

%Y Cf. A062759, A062760, A007947, A003557, A051904, A005117, A001597, A072774.

%K nonn

%O 1,1

%A _Labos Elemer_, Jul 18 2001

%E Offset corrected by _Charles R Greathouse IV_, Sep 18 2015