A378168 a(n) is the number of squares <= 10^n that are not higher powers, i.e., terms of A076467.

2, 6, 24, 87, 292, 959, 3089, 9875, 31410, 99633, 315589, 998889, 3160340, 9996605, 31616816, 99989509, 316209268, 999967330, 3162219896, 9999897769, 31622595517, 99999679010, 316227196708, 999998989804, 3162275866962, 9999996815862, 31622770946248, 99999989953079

Offset: 1

Author

Hugo Pfoertner, Nov 20 2024

Keywords

• nonn

Examples

			a(1) = 2: squares <= 10 are 2^2 and 3^2;
a(2) = 6: 2 squares <= 10 and 5^2, 6^2, 7^2, 10^2, but not 4^2=2^4, 8^2=2^6, and 9^2=3^4;
a(3) = 24: 6 squares <= 100 and all squares between 11^2 and 31^2, except for 16^2=2^8, 25^2=5^4, and 27^2=3^6.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A000290, A001597, A070428, A076467, A089579.

Programs

Mathematica

Table[Sum[MoebiusMu[k]*Floor[10^(n/(2k))-1],{k,Floor[Log2[10^n]-1]}],{n,28}] (* James C. McMahon, Nov 21 2024 *)

Python

from math import gcd
from sympy import integer_nthroot, mobius
def A378168(n): return sum(mobius(k)*(integer_nthroot(10**(n//(a:=gcd(n,b:=k<<1))), b//a)[0]-1) for k in range(1, (10**n).bit_length()-1)) # Chai Wah Wu, Nov 20 2024

Formula

a(n) = Sum_{k=1..floor(log_2(10^n)-1)} mu(k)*floor(10^(n/(2k))-1). - Chai Wah Wu, Nov 20 2024

Extensions

a(20) onwards from Chai Wah Wu, Nov 20 2024