A333645 - OEIS

%I #11 Nov 12 2022 05:25:30

%S 1,2,3,5,5,6,7,12,11,10,11,15,13,14,15,27,17,22,19,25,21,22,23,36,29,

%T 26,37,35,29,30,31,58,33,34,35,55,37,38,39,60,41,42,43,55,55,46,47,81,

%U 55,58,51,65,53,74,55,84,57,58,59,75,61,62,77,121,65,66,67,85,69,70

%N a(n) = Sum_{d|n} uphi(d).

%C Inverse Moebius transform of A047994.

%H Amiram Eldar, <a href="/A333645/b333645.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} uphi(k) * x^k / (1 - x^k).

%F a(n) = Sum_{d|n} A023900(d) * A062949(n/d).

%F From _Amiram Eldar_, Nov 12 2022: (Start)

%F Multiplicative with a(p^e) = (p^(e+1) - e*p + e - 1)/(p-1).

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - (2*p-1)/p^3) = A152649 * A065464 = 0.5793804872... . (End)

%t uphi[1] = 1; uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[n_] := Sum[uphi[d], {d, Divisors[n]}]; Table[a[n], {n, 70}]

%t A023900[n_] := Sum[MoebiusMu[d] d, {d, Divisors[n]}]; A062949[n_] := Sum[EulerPhi[d] DivisorSigma[0, d], {d, Divisors[n]}]; a[n_] := Sum[A023900[d] A062949[n/d], {d, Divisors[n]}]; Table[a[n], {n, 70}]

%t f[p_,e_] := (p^(e+1) - e*p + e - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100] (* _Amiram Eldar_, Nov 12 2022 *)

%o (PARI) uphi(n)=my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); \\ A047994

%o a(n) = sumdiv(n, d, uphi(d)); \\ _Michel Marcus_, Mar 31 2020

%Y Cf. A005117 (fixed points), A023900, A047994, A055653, A062949, A065464, A152649.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Mar 31 2020