%I #50 Mar 29 2021 21:03:34

%S 1,5,9,17,25,41,49,73,89,113,129,169,185,233,257,289,321,385,409,481,

%T 513,561,601,689,721,801,849,921,969,1081,1113,1233,1297,1377,1441,

%U 1537,1585,1729,1801,1897,1961,2121,2169,2337,2417,2513,2601,2785,2849,3017

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

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

%C 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.

%C From _Robert Price_, May 10 2013: (Start)

%C List of sequences that address these extensions:

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

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

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

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

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

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

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

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

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

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

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

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

%C (End)

%C This sequence (A100449) without the GCD qualifier results in A001844. - _Robert Price_, Jun 04 2013

%H Alois P. Heinz, <a href="/A100449/b100449.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = 1 + 4*Sum(phi(k), k=1..n) = 1 + 4*A002088(n). - _Vladeta Jovovic_, Nov 25 2004

%p 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;

%p # second Maple program:

%p b:= proc(n) b(n):= numtheory[phi](n)+`if`(n=0, 0, b(n-1)) end:

%p a:= n-> 1+4*b(n):

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Mar 01 2013

%t 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 *)

%o (PARI) a(n) = 1+4*sum(k=1, n, eulerphi(k) ); \\ _Joerg Arndt_, May 10 2013

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A100449(n):

%o if n == 0:

%o return 1

%o c, j = 0, 2

%o k1 = n//j

%o while k1 > 1:

%o j2 = n//k1 + 1

%o c += (j2-j)*((A100449(k1)-3)//2)

%o j, k1 = j2, n//j2

%o return 2*(n*(n-1)-c+j)+1 # _Chai Wah Wu_, Mar 29 2021

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

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Nov 21 2004

%E More terms from _Vladeta Jovovic_, Nov 25 2004