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

1,1

Links

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

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A048120 A322198 A003759 * A217527 A293774 A226037

Adjacent sequences: A378165 A378166 A378167 * A378169 A378170 A378171

Keyword

nonn

Author

Hugo Pfoertner, Nov 20 2024

Extensions

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

Status

approved