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A209295
Antidiagonal sums of the gcd(.,.) array A109004.
5
0, 2, 5, 8, 12, 14, 21, 20, 28, 30, 37, 32, 52, 38, 53, 60, 64, 50, 81, 56, 92, 86, 85, 68, 124, 90, 101, 108, 132, 86, 165, 92, 144, 138, 133, 152, 204, 110, 149, 164, 220, 122, 237, 128, 212, 234, 181, 140, 288, 182, 245, 216, 252, 158, 297, 244
OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 5000 terms from G. C. Greubel)
FORMULA
a(0) = 0; a(n) = A018804(n) + n for n > 0. [Amended by Georg Fischer, Jan 25 2020]
a(n) = Sum_{d|n} phi(d)*(n/d + 1) for n >= 1. - Peter Luschny, Aug 25 2019
MAPLE
a:= n-> add(igcd(j, n-j), j=0..n):
seq(a(n), n=0..70); # Alois P. Heinz, Aug 25 2019
# Alternative: computes [a(n), n=0..10000] about 25 times faster:
a := n -> add(numtheory:-phi(d)*(n/d + 1), d = numtheory:-divisors(n)):
seq(a(n), n = 0..57); # Peter Luschny, Aug 25 2019
MATHEMATICA
Table[Sum[GCD[n-k, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)
f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := n + Times @@ f @@@ FactorInteger[n]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Apr 28 2023 *)
PROG
(PARI) a(n) = n + sum(k=1, n, gcd(n, k)); \\ Michel Marcus, Jan 05 2018
(Magma)
A209295:= func< n | n eq 0 select 0 else (&+[(n/d+1)*EulerPhi(d): d in Divisors(n)]) >;
[A209295(n): n in [0..40]]; // G. C. Greubel, Jun 24 2024
(SageMath)
def A209295(n): return sum((n/k+1)*euler_phi(k) for k in (1..n) if (k).divides(n))
[A209295(n) for n in range(41)] # G. C. Greubel, Jun 24 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Jan 17 2013
STATUS
approved