

A100449


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


13



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
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OFFSET

0,2


COMMENTS

Note that GCD(0,m) = m for any m.
From Robert Price, May 10 2013: (Start)
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..1000


FORMULA

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


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(n1)) 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 *)


PROG

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


CROSSREFS

Cf. A018805, A100448, A100450, A027430, etc.
Sequence in context: A182388 A080335 A089109 * A146284 A175543 A200078
Adjacent sequences: A100446 A100447 A100448 * A100450 A100451 A100452


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Nov 21 2004


EXTENSIONS

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.
More terms from Vladeta Jovovic, Nov 25 2004
See the Comments section for sequences that address these extensions.  Robert Price, May 10 2013


STATUS

approved



