A062761 Number of powerful numbers between 2^(n-1)+1 and 2^n.

1, 0, 1, 1, 2, 3, 3, 7, 8, 12, 17, 25, 36, 50, 74, 105, 152, 213, 306, 437, 620, 882, 1256, 1781, 2531, 3588, 5091, 7221, 10225, 14504, 20542, 29101, 41214, 58369, 82638, 116986, 165610, 234387, 331738, 469429, 664291, 939924, 1329876, 1881500, 2661826, 3765629

Offset: 0

Author

Labos Elemer, Jul 16 2001

Keywords

• nonn

Examples

			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.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..127 (terms 0..90 from Daniel Suteu)

Crossrefs

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

Programs

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

PARI

Q(n) = my(s=0); forsquarefree(k=1, sqrtnint(n, 3), s += sqrtint(n\k[1]^3)); s;
a(n) = if(n==0, 1, Q(2^n) - Q(2^(n-1))); \\ Daniel Suteu, Feb 18 2020

Python

# uses code from A062762
def A062761(n): return A062762(n)-A062762(n-1) if n else 1 # Chai Wah Wu, Sep 13 2024

Formula

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

Sum_{k=0..n} a(k) = A062762(n). - Daniel Suteu, Feb 18 2020

Extensions

a(19)-a(29) from Daniel Suteu, Aug 25 2019

a(30)-a(45) from Daniel Suteu, Feb 18 2020