A387980 Sum of squarefree numbers <= 10^n.

1, 34, 2967, 303076, 30420034, 3039711199, 303961062910, 30396557311887, 3039633904822886, 303963567619632057, 30396354343039613622, 3039635516178239919779, 303963551353876732927386, 30396355090144154315002969, 3039635509323573017710407946, 303963550927241195239795351754

Offset

0,2

Comments

Asymptotically ~ 0.30396355...*(10^n)^2 = (3/Pi^2)*(10^n)^2. Curiously, this limit is the same as A064018.

Links

Henri Lifchitz, Table of n, a(n) for n = 0..24 (terms 21..22 from Chai Wah Wu), Jan 10 2026

Formula

a(n) = (10^n)*(10^n+1)/2 - A388005(n).

Example

The first squarefree numbers are: 1, 2, 3, 5, 6, 7, 10, ...
So, a(1) = 1+2+3+5+6+7+10 = 34.

Mathematica

SumSquareFree[n_Integer?Positive] := Total@Select[Range[n], SquareFreeQ]; Table[ SumSquareFree[10^n], {n, 0, 6}]

Prog

(PARI) a(n) = my(s=0); forsquarefree(x=1, 10^n, s+=x[1]); s; \\ Michel Marcus, Oct 15 2025
(Python)
from math import isqrt, comb
from sympy import mobius
def A387980(n): return sum(mobius(k)*(s:=k**2)*comb(10**n//s+1, 2) for k in range(1, isqrt(10**n)+1)) # Chai Wah Wu, Oct 20 2025

Crossrefs

Cf. A005117, A388005, A143658, A173143 (partial sums).

Sequence in context: A160471 A252709 A138590 * A069223 A258499 A218718

Adjacent sequences: A387977 A387978 A387979 * A387981 A387982 A387983

Keyword

nonn,hard

Author

Henri Lifchitz, Oct 13 2025

Status

approved