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

Author

Vaclav Kotesovec, Sep 10 2018

Keywords

• nonn

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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