A319087 a(n) = Sum_{k=1..n} k^2*phi(k), where phi is the Euler totient function A000010.

1, 5, 23, 55, 155, 227, 521, 777, 1263, 1663, 2873, 3449, 5477, 6653, 8453, 10501, 15125, 17069, 23567, 26767, 32059, 36899, 48537, 53145, 65645, 73757, 86879, 96287, 119835, 127035, 155865, 172249, 194029, 212525, 241925, 257477, 306761, 332753, 369257

Offset

1,2

Comments

Comment from N. J. A. Sloane, Mar 22 2020: (Start)

Theorem: Sum_{ 1<=i<=n, 1<=j<=n, gcd(i,j)=1 } i*j = a(n).

Proof: From the Apostol reference we know that:

Sum_{ 1<=i<=n, gcd(i,n)=1 } i = n*phi(n)/2 (see A023896).

We use induction on n. The result is true for n=1.

Then a(n) - a(n-1) = 2*Sum_{ i=1..n-1, gcd(i,n)=1 } n*i = n^2*phi(n). QED (End)

References

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 16, the function phi_1(n).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) ~ 3*n^4 / (2*Pi^2).

Mathematica

Accumulate[Table[k^2*EulerPhi[k], {k, 1, 50}]]

Prog

(PARI) a(n) = sum(k=1, n, k^2*eulerphi(k)); \\ Michel Marcus, Sep 12 2018

Crossrefs

Cf. A000010, A002088, A011755, A023896, A053191.

Sequence in context: A053664 A186030 A092544 * A098499 A075565 A075707

Adjacent sequences: A319084 A319085 A319086 * A319088 A319089 A319090

Keyword

nonn

Author

Vaclav Kotesovec, Sep 10 2018

Status

approved