

A127416


a(n) = Sum_{1<=k<=n, gcd(k,n)=1} (Sum_{i=1..k} gcd(i, k)).


1



1, 1, 4, 6, 17, 10, 41, 28, 54, 40, 122, 44, 183, 82, 130, 140, 340, 102, 436, 156, 304, 254, 673, 184, 665, 360, 604, 388, 1143, 232, 1335, 584, 854, 656, 1138, 484, 1997, 842, 1238, 768, 2486, 508, 2762, 1044, 1382, 1314, 3339, 816, 3139, 1160, 2276, 1600
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OFFSET

1,3


COMMENTS

a(n) = Sum_{1<=k<=n, gcd(k,n)=1} A018804(k) where A018804(n) = Sum_{k=1..n} gcd(k,n).


LINKS



FORMULA

M * V where M = A054521 is an infinite lower triangular matrix and V = A018804 is a vector: (1, 3, 5, 8, 9, 15, 13, ...).


EXAMPLE

a(6) = 10 since the relative primes of 6 are 1 and 5, A018804(1) + A018804(5) = 1 + 9 = 10.


MATHEMATICA

f[p_, e_] := (e*(p  1)/p + 1)*p^e; pil[n_] := Times @@ (f @@@ FactorInteger[n]); a[n_] := Sum[If[GCD[n, k] == 1, pil[k], 0], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, Jul 19 2019 *)


PROG

(PARI) f(n) = sum(i=1, n, gcd(n, i)); \\ A018804
a(n) = sum(k=1, n, if (gcd(k, n) == 1, f(k))); \\ Michel Marcus, Jul 19 2019


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



