A279911 - OEIS

%I #30 May 25 2023 08:05:36

%S 0,1,2,4,6,11,10,22,22,31,28,56,36,79,58,72,86,137,80,172,112,145,148,

%T 254,146,261,208,274,230,407,182,466,342,375,360,448,322,667,456,528,

%U 444,821,384,904,592,635,676,1082,574,1051,692,924,836,1379,732,1154,912,1153

%N a(n) = Sum_{i=1..n} denominator(n^i/i).

%H Daniel Suteu, <a href="/A279911/b279911.txt">Table of n, a(n) for n = 0..10000</a>

%F From _Daniel Suteu_, Jul 28 2019: (Start)

%F a(prime(n)) = A072205(n).

%F a(p^k) = (p^(2*k+1) + p + 2) / (2*(p+1)), for prime powers p^k.

%F a(n) = Sum_{k=1..n} gcd(m, k), where m = A095996(n).

%F a(n) = Sum_{k=1..n} f(n,k), where f(n,k) is the largest divisor d of k for which gcd(d, n) = 1. (End)

%F a(n) = Sum_{1<=k<=n, gcd(n,k)=1} phi(k)*floor(n/k). - _Ridouane Oudra_, May 24 2023

%p A279911:=n->add(denom(n^i/i), i=1..n): seq(A279911(n), n=0..100);

%o (PARI) a(n) = sum(k=1, n, if(gcd(n,k) == 1, k, denominator(n^k/k))); \\ _Daniel Suteu_, Jul 28 2019

%o (PARI) a(n) = sum(k=1, n, if(gcd(n,k) == 1, k, vecmax(select(d->gcd(d, n) == 1, divisors(k))))); \\ _Daniel Suteu_, Jul 28 2019

%o (PARI) a(n) = my(f=factor(n)[,1]); sum(k=1, n, if(gcd(n, k) == 1, k, gcd(vector(#f, j, k / f[j]^valuation(k, f[j]))))); \\ _Daniel Suteu_, Jul 29 2019

%o (Magma) [0] cat [&+[Denominator(n^i/i):i in [1..n]]:n in [1..60]]; // _Marius A. Burtea_, Jul 29 2019

%Y Cf. A279912.

%K nonn,easy

%O 0,3

%A _Wesley Ivan Hurt_, Dec 22 2016