A072292 - OEIS

%I #19 Nov 14 2022 17:17:00

%S 0,0,0,1,1,1,1,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,7,7,7,

%T 7,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,

%U 10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11

%N Number of proper powers b^d <= n (b > 1, d > 1).

%C Base b = 1 is excluded since 1 would be 1^d for any degree d (degree of power not well defined).

%H Reinhard Zumkeller, <a href="/A072292/b072292.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>.

%t a[n_] := (pp = Reap[ Do[ If[b^d <= n, Sow[b^d]], {b, 2, Sqrt[n]}, {d, 2, Log[2, n]}]]; If[pp == {Null, {}}, 0, Length[ Union[ pp[[2, 1]]]]]); Table[a[n], {n, 1, 90}](* _Jean-François Alcover_, May 16 2012 *)

%t Module[{nn=10,pp},pp=Union[Flatten[Table[a^b,{a,2,nn},{b,2,nn}]]];Accumulate[ Table[ If[ MemberQ[pp,n],1,0],{n,2^nn}]]] (* _Harvey P. Dale_, Nov 14 2022 *)

%o (PARI) A072292(n)=n=floor(n)+.5;-sum(k=2,log(n)\log(2),floor(n^(1/k)-1)*moebius(k))

%o \\ _Charles R Greathouse IV_, Sep 07 2010

%o (Haskell)

%o a072292 n = a072292_list !! (n-1)

%o a072292_list = scanl (+) 0 $ tail a075802_list

%o -- _Reinhard Zumkeller_, May 26 2012

%Y a(i)=A069637(i) for i<36=6^2. Cf. A001597.

%Y Cf. A075802 (first differences).

%K nonn,easy,nice

%O 1,8

%A _Reinhard Zumkeller_, Jul 12 2002

%E Edited by _Daniel Forgues_, Mar 03 2009