cp's OEIS Frontend

A377435 Number of perfect-powers x in the range 2^n <= x < 2^(n+1).

%I A377435 #14 Nov 06 2024 10:19:50
%S A377435 1,0,1,2,3,3,5,7,8,11,16,24,32,42,61,82,118,166,231,322,453,635,892,
%T A377435 1253,1767,2487,3505,4936,6959,9816,13850,19538,27578,38933,54972,
%U A377435 77641,109668,154922,218879,309277,437047,617658,872968,1233896,1744153,2465547,3485478
%N A377435 Number of perfect-powers x in the range 2^n <= x < 2^(n+1).
%C A377435 Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.
%C A377435 Also the number of perfect-powers with n bits.
%F A377435 For n != 1, a(n) = A377467(n) + 1.
%e A377435 The perfect-powers in each prescribed range (rows):
%e A377435     1
%e A377435     .
%e A377435     4
%e A377435     8    9
%e A377435    16   25   27
%e A377435    32   36   49
%e A377435    64   81  100  121  125
%e A377435   128  144  169  196  216  225  243
%e A377435   256  289  324  343  361  400  441  484
%e A377435   512  529  576  625  676  729  784  841  900  961 1000
%e A377435 Their binary expansions (columns):
%e A377435   1  .  100  1000  10000  100000  1000000  10000000  100000000
%e A377435              1001  11001  100100  1010001  10010000  100100001
%e A377435                    11011  110001  1100100  10101001  101000100
%e A377435                                   1111001  11000100  101010111
%e A377435                                   1111101  11011000  101101001
%e A377435                                            11100001  110010000
%e A377435                                            11110011  110111001
%e A377435                                                      111100100
%t A377435 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A377435 Table[Length[Select[Range[2^n,2^(n+1)-1],perpowQ]],{n,0,15}]
%o A377435 (Python)
%o A377435 from sympy import mobius, integer_nthroot
%o A377435 def A377435(n):
%o A377435     if n==0: return 1
%o A377435     def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
%o A377435     return f((1<<n+1)-1)-f((1<<n)-1) # _Chai Wah Wu_, Nov 05 2024
%Y A377435 The union of all numbers counted is A001597, without powers of two A377702.
%Y A377435 The version for squarefree numbers is A077643.
%Y A377435 These are the first differences of A188951.
%Y A377435 The version for prime-powers is A244508.
%Y A377435 For primes instead of powers of 2 we have A377432, zeros A377436.
%Y A377435 Not counting powers of 2 gives A377467.
%Y A377435 The version for non-perfect-powers is A377701.
%Y A377435 A000040 lists the primes, differences A001223.
%Y A377435 A000961 lists the powers of primes, differences A057820.
%Y A377435 A001597 lists the perfect-powers, differences A053289.
%Y A377435 A007916 lists the non-perfect-powers, differences A375706.
%Y A377435 A081676 gives the greatest perfect-power <= n.
%Y A377435 A131605 lists perfect-powers that are not prime-powers.
%Y A377435 A377468 gives the least perfect-power > n.
%Y A377435 Cf. A000015, A013597, A014210, A014234, A023055, A031218, A045542, A052410, A065514, A069623, A216765, A345531, A377434.
%K A377435 nonn
%O A377435 0,4
%A A377435 _Gus Wiseman_, Nov 04 2024
%E A377435 a(26)-a(46) from _Chai Wah Wu_, Nov 05 2024