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 A319087 a(n) = Sum_{k=1..n} k^2*phi(k), where phi is the Euler totient function A000010. 8
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified May 28 01:34 EDT 2024. Contains 372900 sequences. (Running on oeis4.)