cp's OEIS Frontend

A377467 Number of perfect-powers x in the range 2^n < x < 2^(n+1).

%I A377467 #16 Nov 06 2024 10:19:47
%S A377467 0,0,0,1,2,2,4,6,7,10,15,23,31,41,60,81,117,165,230,321,452,634,891,
%T A377467 1252,1766,2486,3504,4935,6958,9815,13849,19537,27577,38932,54971,
%U A377467 77640,109667,154921,218878,309276,437046,617657,872967,1233895,1744152,2465546,3485477
%N A377467 Number of perfect-powers x in the range 2^n < x < 2^(n+1).
%C A377467 Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.
%C A377467 Also the number of perfect-powers, except for powers of 2, with n bits.
%F A377467 For n != 1, a(n) = A377435(n) - 1.
%e A377467 The perfect-powers in each prescribed range (rows):
%e A377467     .
%e A377467     .
%e A377467     .
%e A377467     9
%e A377467    25   27
%e A377467    36   49
%e A377467    81  100  121  125
%e A377467   144  169  196  216  225  243
%e A377467   289  324  343  361  400  441  484
%e A377467   529  576  625  676  729  784  841  900  961 1000
%e A377467 The binary expansions for n >= 3 (columns):
%e A377467     1001  11001  100100  1010001  10010000  100100001
%e A377467           11011  110001  1100100  10101001  101000100
%e A377467                          1111001  11000100  101010111
%e A377467                          1111101  11011000  101101001
%e A377467                                   11100001  110010000
%e A377467                                   11110011  110111001
%e A377467                                             111100100
%t A377467 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A377467 Table[Length[Select[Range[2^n+1,2^(n+1)-1],perpowQ]],{n,0,15}]
%o A377467 (Python)
%o A377467 from sympy import mobius, integer_nthroot
%o A377467 def A377467(n):
%o A377467     def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
%o A377467     return f((1<<n+1)-1)-f((1<<n)) # _Chai Wah Wu_, Nov 05 2024
%Y A377467 The version for squarefree numbers is A077643.
%Y A377467 The version for prime-powers is A244508.
%Y A377467 For primes instead of powers of 2 we have A377432, zeros A377436.
%Y A377467 Including powers of 2 in the range gives A377435.
%Y A377467 The version for non-perfect-powers is A377701.
%Y A377467 The union of all numbers counted is A377702.
%Y A377467 A000040 lists the primes, differences A001223.
%Y A377467 A000961 lists the powers of primes, differences A057820.
%Y A377467 A001597 lists the perfect-powers, differences A053289.
%Y A377467 A007916 lists the non-perfect-powers, differences A375706.
%Y A377467 A081676 gives the greatest perfect-power <= n.
%Y A377467 A131605 lists perfect-powers that are not prime-powers.
%Y A377467 A377468 gives the least perfect-power > n.
%Y A377467 Cf. A000015, A013597, A014210, A014234, A023055, A031218, A045542, A052410, A065514, A069623, A216765, A246655, A345531.
%K A377467 nonn
%O A377467 0,5
%A A377467 _Gus Wiseman_, Nov 04 2024
%E A377467 a(26)-a(46) from _Chai Wah Wu_, Nov 05 2024