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A062770 n/[largest power of squarefree kernel] equals 1; perfect powers of sqf-kernels (or sqf-numbers).

%I A062770 #33 Aug 10 2025 07:07:49
%S A062770 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 A062770 32,33,34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,
%U A062770 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 A062770 n/[largest power of squarefree kernel] equals 1; perfect powers of sqf-kernels (or sqf-numbers).
%C A062770 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 A062770 Michael De Vlieger, <a href="/A062770/b062770.txt">Table of n, a(n) for n = 1..10000</a>
%F A062770 A062760(a(n)) = 1, i.e., a(n)/(A007947(a(n))^A051904(a(n))) = a(n)/A062759(a(n)) = 1.
%F A062770 a(n) = A072774(n+1). - _Chai Wah Wu_, Aug 19 2024
%e A062770 Primes, squarefree numbers and perfect powers are here.
%e A062770 From _Michael De Vlieger_, Jun 24 2022 (Start):
%e A062770 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 A062770 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 A062770 Select[Range[2, 2^16], Length@ Union@ FactorInteger[#][[All, -1]] == 1 &] (* _Michael De Vlieger_, Jun 24 2022 *)
%o A062770 (PARI) is(n)=ispower(n,,&n); issquarefree(n) && n>1 \\ _Charles R Greathouse IV_, Sep 18 2015
%o A062770 (PARI) is(n)=#Set(factor(n)[,2])==1 \\ _Charles R Greathouse IV_, Sep 18 2015
%o A062770 (Python)
%o A062770 from math import isqrt
%o A062770 from sympy import mobius, integer_nthroot
%o A062770 def A062770(n):
%o A062770     def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
%o A062770     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 A062770     kmin, kmax = 1,2
%o A062770     while f(kmax) >= kmax:
%o A062770         kmax <<= 1
%o A062770     while True:
%o A062770         kmid = kmax+kmin>>1
%o A062770         if f(kmid) < kmid:
%o A062770             kmax = kmid
%o A062770         else:
%o A062770             kmin = kmid
%o A062770         if kmax-kmin <= 1:
%o A062770             break
%o A062770     return kmax # _Chai Wah Wu_, Aug 19 2024
%Y A062770 Cf. A062759, A062760, A007947, A003557, A051904, A005117, A001597, A072774.
%K A062770 nonn
%O A062770 1,1
%A A062770 _Labos Elemer_, Jul 18 2001
%E A062770 Offset corrected by _Charles R Greathouse IV_, Sep 18 2015