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A174405 Partial sums of Sum_{k=1..n} n/gcd(n,k), or partial sums of Sum_{d|n} d*phi(d) (see A057660). 1
1, 4, 11, 22, 43, 64, 107, 150, 211, 274, 385, 462, 619, 748, 895, 1066, 1339, 1522, 1865, 2096, 2397, 2730, 3237, 3538, 4059, 4530, 5077, 5550, 6363, 6804, 7735, 8418, 9195, 10014, 10917, 11588, 12921, 13950, 15049, 15952, 17593, 18496, 20303, 21524, 22805, 24326, 26489, 27686, 29787, 31350, 33261, 34988 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..52.

Marko Riedel, answer to 'Euler phi function, number theory' (2014)

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 gives a(n) ~ 2/Pi^2 * zeta(3) * n^3. - Charles R Greathouse IV, May 21 2014

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.

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

Cf. A000010, A057660.

Sequence in context: A295957 A259574 A008249 * A016274 A092656 A269743

Adjacent sequences:  A174402 A174403 A174404 * A174406 A174407 A174408

KEYWORD

nonn

AUTHOR

Jonathan Vos Post, Nov 27 2010

STATUS

approved

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Last modified July 21 06:55 EDT 2019. Contains 325192 sequences. (Running on oeis4.)