A069194 a(n) = Sum_{d|n} (n/d)*phi(n)/phi(d).
1, 3, 7, 13, 21, 21, 43, 53, 64, 63, 111, 91, 157, 129, 147, 213, 273, 192, 343, 273, 301, 333, 507, 371, 526, 471, 577, 559, 813, 441, 931, 853, 777, 819, 903, 832, 1333, 1029, 1099, 1113, 1641, 903, 1807, 1443, 1344, 1521, 2163, 1491, 2108, 1578, 1911, 2041
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Magma
[&+[(n div d)*EulerPhi(n) div EulerPhi(d):d in Divisors(n)]:n in [1..52]]; // Marius A. Burtea, Sep 15 2019
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Maple
for i from 1 to 100 do d := divisors(i): a[i] := i*phi(i)*sum(1/d[j]/phi(d[j]),j=1..nops(d)) od:seq(a[j],j=1..100);
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Mathematica
f[p_, e_] := p^e*(p^e - p^(e-1)) + (p^(2*e) - 1)/(p^2 - 1) ; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2019 *)
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PARI
a(n) = sumdiv(n, d, n/d*eulerphi(n)/eulerphi(d)); \\ Michel Marcus, Sep 15 2019
Formula
Multiplicative with a(p^e) = p^e*(p^e - p^(e-1)) + (p^(2*e) - 1)/(p^2 - 1). - Amiram Eldar, Sep 15 2019
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 1/p^2 + 1/p^5) = 0.2550149528... . - Amiram Eldar, Oct 28 2022
Extensions
More terms from Sascha Kurz, Jan 03 2003