OFFSET
1,2
COMMENTS
The subsequence of primes in this sequence begins 11, 43, 107, 211, 619, 5077, 26489. The subsequence of squares in this sequence begins 1, 4, 64, 18496 = 2^6 * 17^2.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Marko Riedel, answer to 'Euler phi function, number theory', MathStackExchange, 2014.
Günter Rote, Moritz Rüber, and Morteza Saghafian, Grid Peeling of Parabolas, arXiv:2402.15787 [cs.CG], 2024. See p. 9.
J. Sándor and A. V. Kramer, Über eine zahlentheoretische Funktion. Mathematica Moravica, 3 (1999), 53-62.
FORMULA
a(n) = Sum_{i=1..n} A057660(i) = Sum_{i=1..n} Sum_{k=1..i} i/gcd(i,k) = Sum_{i=1..n} ( Sum_{d|i} A000010(d^2) ) = Sum_{i=1..n} ( Sum_{d|i} d*A000010(d) ) = Sum_{i=1..n} (sum of the orders of the elements in a cyclic group with i elements).
Riedel, and Sándor&Kramer give a(n) ~ 2/Pi^2 * zeta(3) * n^3. - Charles R Greathouse IV, May 21 2014; Günter Rote, November 3 2021
G.f.: (1/(1 - x))*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, Aug 31 2017
a(n) = Sum_{k=1..n} k * phi(k) * floor(n/k), where phi(k) is the Euler totient function. - Daniel Suteu, May 30 2018
EXAMPLE
a(9) = 1 + 3 + 7 + 11 + 21 + 21 + 43 + 43 + 61 = 211 is prime.
MATHEMATICA
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Accumulate @ Array[a, 100] (* Amiram Eldar, Nov 21 2020 *)
PROG
(PARI) a(n)=sum(k=1, n, sumdiv(k, d, eulerphi(d)*d)) \\ Charles R Greathouse IV, May 21 2014
(PARI) a(n) = sum(k=1, n, k * eulerphi(k) * (n\k)); \\ Michel Marcus, May 30 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Nov 27 2010
STATUS
approved