A100449 Number of ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.

1, 5, 9, 17, 25, 41, 49, 73, 89, 113, 129, 169, 185, 233, 257, 289, 321, 385, 409, 481, 513, 561, 601, 689, 721, 801, 849, 921, 969, 1081, 1113, 1233, 1297, 1377, 1441, 1537, 1585, 1729, 1801, 1897, 1961, 2121, 2169, 2337, 2417, 2513, 2601, 2785, 2849, 3017

Offset: 0

Author

N. J. A. Sloane, Nov 21 2004

Keywords

• nonn

Comments

Note that gcd(0,m) = m for any m.

I would also like to get the sequences of the numbers of distinct sums i+j (also distinct products i*j) over all ordered pairs (i,j) with |i| + |j| <= n; also over all ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.

From Robert Price, May 10 2013: (Start)

List of sequences that address these extensions:

Distinct sums i+j with or without the GCD qualifier results in a(n)=2n+1 (A005408).

Distinct products i*j without the GCD qualifier is given by A225523.

Distinct products i*j with the GCD qualifier is given by A225526.

With the restriction i,j >= 0 ...

Distinct sums or products equal to n is trivial and always equals one (A000012).

Distinct sums <=n with or without the GCD qualifier results in a(n)=n (A001477).

Distinct products <=n without the GCD qualifier is given by A225527.

Distinct products <=n with the GCD qualifier is given by A225529.

Ordered pairs with the sum = n without the GCD qualifier is a(n)=n+1.

Ordered pairs with the sum = n with the GCD qualifier is A225530.

Ordered pairs with the sum <=n without the GCD qualifier is A000217(n+1).

Ordered pairs with the sum <=n with the GCD qualifier is A225531.

(End)

This sequence (A100449) without the GCD qualifier results in A001844. - Robert Price, Jun 04 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A018805, A100448, A100450, A027430, etc.

Programs

Maple

f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from -n to n do for j from -n to n do if abs(i) + abs(j) <= n then t2:=gcd(i,j); if t2 <= 1 then t1:=t1+1; fi; fi; od: od: t1; end;
# second Maple program:
b:= proc(n) b(n):= numtheory[phi](n)+`if`(n=0, 0, b(n-1)) end:
a:= n-> 1+4*b(n):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 01 2013

Mathematica

f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] <= n && GCD[i, j] <= 1, {i, j}, {0, 0}], {i, -n, n}, {j, -n, n}], 1]]]; Table[ f[n], {n, 0, 49}] (* Robert G. Wilson v, Dec 14 2004 *)

PARI

a(n) = 1+4*sum(k=1, n, eulerphi(k) ); \\ Joerg Arndt, May 10 2013

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A100449(n):
    if n == 0:
        return 1
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*((A100449(k1)-3)//2)
        j, k1 = j2, n//j2
    return 2*(n*(n-1)-c+j)+1 # Chai Wah Wu, Mar 29 2021

Formula

a(n) = 1 + 4*Sum(phi(k), k=1..n) = 1 + 4*A002088(n). - Vladeta Jovovic, Nov 25 2004

Extensions

More terms from Vladeta Jovovic, Nov 25 2004