A057828 Number of perfect squares, k^2, where k^2 <= n and gcd(k,n) = 1.

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 1, 4, 2, 3, 2, 4, 1, 4, 3, 4, 3, 5, 1, 5, 3, 4, 3, 4, 2, 6, 3, 4, 2, 6, 2, 6, 3, 3, 3, 6, 2, 6, 3, 5, 4, 7, 3, 6, 3, 5, 4, 7, 2, 7, 4, 4, 4, 7, 3, 8, 4, 6, 2, 8, 3, 8, 4, 5, 4, 7, 3, 8, 3, 6, 5, 9, 2, 8, 5, 6, 5, 9, 2, 8, 5, 6, 5, 8, 3, 9, 4, 6, 4, 10, 3, 10, 5

Offset

1,5

Comments

Number of square totatives of n, i.e., number of perfect squares less than n that are coprime to n. - Michael De Vlieger, Dec 11 2017

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Totative.

Formula

a(n) = Sum_{d|n} mu(d)*floor(sqrt(n)/d). - Ridouane Oudra, Jan 26 2025

a(n) = Sum_{k=1..floor(sqrt(n))} A054521(n,k). - Ridouane Oudra, Mar 25 2025

Example

Only 2 squares, 1 and 9, are <= 14 and relatively prime to 14. So a(14) = 2.

Maple

with(numtheory): seq(add(mobius(d)*floor(sqrt(n)/d), d in divisors(n)), n=1..100); # Ridouane Oudra, Jan 26 2025

Mathematica

Table[Count[Range[Sqrt@ n]^2, _?(CoprimeQ[#, n] &)], {n, 104}]

Prog

(Haskell)
a057828 x = length $ filter ((== 1) . (gcd x)) $
                     takeWhile (<= x) $ tail a000290_list
-- Reinhard Zumkeller, Jul 22 2012
(PARI) a(n) = sumdiv(n, d, moebius(d)*(sqrt(n)\d)); \\ Michel Marcus, Jan 27 2025
(PARI) a(n, f=factor(n))=my(g=f, d); g[, 2]=vectorv(#f~, i, 1); d=divisors(g, 1); sum(i=1, #d, moebius(d[i][2])*sqrtint(n\d[i][1]^2)) \\ Charles R Greathouse IV, Mar 26 2025

Crossrefs

Cf. A000290, A010051, A054521.

Sequence in context: A322873 A332897 A026904 * A082498 A112223 A324810

Adjacent sequences: A057825 A057826 A057827 * A057829 A057830 A057831

Keyword

nonn,look

Author

Leroy Quet, Nov 08 2000

Status

approved