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Number of powerful numbers between 2^(n-1)+1 and 2^n.
2

%I #35 Sep 16 2024 12:47:22

%S 1,0,1,1,2,3,3,7,8,12,17,25,36,50,74,105,152,213,306,437,620,882,1256,

%T 1781,2531,3588,5091,7221,10225,14504,20542,29101,41214,58369,82638,

%U 116986,165610,234387,331738,469429,664291,939924,1329876,1881500,2661826,3765629

%N Number of powerful numbers between 2^(n-1)+1 and 2^n.

%H Chai Wah Wu, <a href="/A062761/b062761.txt">Table of n, a(n) for n = 0..127</a> (terms 0..90 from Daniel Suteu)

%F Number of terms x from A001694 for which A029837(x)=n.

%F Sum_{k=0..n} a(k) = A062762(n). - _Daniel Suteu_, Feb 18 2020

%e 64 < {72,81,100,108,121,125,128} <= 128, i.e., 7 powerful numbers are between 2^6 and 2^7, so a(7)=7.

%o (PARI) a(n) = my(ka = if (n==0, 1, 2^(n-1)+1)); #select(x->ispowerful(x), [ka..2^n]); \\ _Michel Marcus_, Aug 25 2019

%o (PARI) Q(n) = my(s=0); forsquarefree(k=1, sqrtnint(n, 3), s += sqrtint(n\k[1]^3)); s;

%o a(n) = if(n==0, 1, Q(2^n) - Q(2^(n-1))); \\ _Daniel Suteu_, Feb 18 2020

%o (Python)

%o # uses code from A062762

%o def A062761(n): return A062762(n)-A062762(n-1) if n else 1 # _Chai Wah Wu_, Sep 13 2024

%Y Cf. A001694, A029837, A036380, A036386, A062762.

%K nonn

%O 0,5

%A _Labos Elemer_, Jul 16 2001

%E a(19)-a(29) from _Daniel Suteu_, Aug 25 2019

%E a(30)-a(45) from _Daniel Suteu_, Feb 18 2020