A046657 a(n) = A002088(n)/2.

1, 2, 3, 5, 6, 9, 11, 14, 16, 21, 23, 29, 32, 36, 40, 48, 51, 60, 64, 70, 75, 86, 90, 100, 106, 115, 121, 135, 139, 154, 162, 172, 180, 192, 198, 216, 225, 237, 245, 265, 271, 292, 302, 314, 325, 348, 356, 377, 387, 403, 415, 441, 450, 470, 482

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) = |{(x,y) : 1 <= x <= y <= n, x + y <= n, 1 = gcd(x,y)}| = |{(x,y) : 1 <= x <= y <= n, x + y > n, 1 = gcd(x,y)}|. - Steve Butler, Mar 31 2006

Brousseau proved that if the starting numbers of a generalized Fibonacci sequence are <= n (for n > 1) then the number of such sequences with relatively prime successive terms is a(n). - Amiram Eldar, Mar 31 2017

Links

• Giovanni Resta, Table of n, a(n) for n = 2..10000
• Alfred Brousseau, A Note on the Number of Fibonacci Sequences, The Fibonacci Quarterly, Vol. 10, No. 6 (1972), pp. 657-658.

Crossrefs

Cf. A002088.

Partial sums of A023022.

Programs

GAP

List([2..60],n->Sum([1..n],k->Phi(k)/2)); # Muniru A Asiru, Mar 05 2018

Maple

a:=n->sum(numtheory[phi](k),k=1..n): seq(a(n)/2, n=2..60); # Muniru A Asiru, Mar 05 2018

Mathematica

Rest@ Accumulate[EulerPhi@ Range@ 56]/2 (* Michael De Vlieger, Apr 02 2017 *)

PARI

a(n) = sum(k=1, n, eulerphi(k))/2; \\ Michel Marcus, Apr 01 2017

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A046657(n): # based on second formula in A018805
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(4*A046657(k1)-1)
        j, k1 = j2, n//j2
    return (n*(n-1)-c+j)//4 # Chai Wah Wu, Mar 25 2021

Formula

a(n) = 1/2 + Sum_{iJoseph Wheat, Feb 22 2018