cp's OEIS Frontend

A299822 Product of Euler's totient and the squarefree kernel, a(n) = phi(n)*rad(n).

%I A299822 #70 Jan 27 2024 05:55:37
%S A299822 1,2,6,4,20,12,42,8,18,40,110,24,156,84,120,16,272,36,342,80,252,220,
%T A299822 506,48,100,312,54,168,812,240,930,32,660,544,840,72,1332,684,936,160,
%U A299822 1640,504,1806,440,360,1012,2162,96,294,200,1632,624,2756,108,2200,336,2052,1624
%N A299822 Product of Euler's totient and the squarefree kernel, a(n) = phi(n)*rad(n).
%C A299822 A permutation of A323333. - _Amiram Eldar_, Sep 19 2020
%H A299822 Amiram Eldar, <a href="/A299822/b299822.txt">Table of n, a(n) for n = 1..10000</a>
%F A299822 a(n) = A000010(n)*A007947(n) = n*A173557(n).
%F A299822 Dirichlet g.f.: zeta(s-1)*Product_{p prime} (1 - 2*p^(1-s) + p^(2-s)), corrected by _Vaclav Kotesovec_, Dec 18 2019
%F A299822 Multiplicative with a(p^e) = p*(p-1)*p^(e-1).
%F A299822 a(n) = n*abs(A023900(n)). (Trivially rephrasing a formula in A173557.) - _Omar E. Pol_, Feb 19 2018
%F A299822 a(2^e) = 2^e. (Special case of above.) - _Omar E. Pol_, Feb 19 2018
%F A299822 A003557(n) | a(n). - _R. J. Mathar_, Feb 26 2018
%F A299822 From _Vaclav Kotesovec_, Dec 18 2019: (Start)
%F A299822 Dirichlet g.f.: zeta(s-1) * zeta(s-2) * Product_{primes p} (1 + 2*p^(3-2*s) - p^(4-2*s) - 2*p^(1-s)).
%F A299822 Sum_{k=1..n} a(k) ~ c * Pi^2 * n^3 / 18, where c = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.2867474284344787341078927... (End)
%F A299822 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p-1)^2) = 2.826419... (A065485). - _Amiram Eldar_, Sep 19 2020
%F A299822 G.f. for a signed version of the sequence: Sum_{n >= 1} mu(n)*n^2*x^n/(1 - x^n)^2 = Sum_{n >= 1} (-1)^omega(n)*a(n)*x^n = x - 2*x^2 - 6*x^3 - 4*x^4 - 20*x^5 + 12*x^6 - 42*x^7 - 8*x^8 - 18*x^9 + 40*x^10 - ..., where mu(n) is the Möbius function A008683(n) and omega(n) = A001221(n) is the number of distinct primes dividing n. - _Peter Bala_, Mar 05 2022
%p A299822 A299822 := proc(n)
%p A299822     local a,p,e,pe;
%p A299822     a := 1;
%p A299822     for pe in ifactors(n)[2] do
%p A299822         p := pe[1] ; e:= pe[2] ;
%p A299822         a := a*p*(p-1)*p^(e-1) ;
%p A299822     end do:
%p A299822     a ;
%p A299822 end proc:
%p A299822 seq(A299822(n),n=1..130) ;
%t A299822 Array[EulerPhi[#] SelectFirst[Reverse@ Divisors@ #, SquareFreeQ] &, 58] (* _Michael De Vlieger_, Feb 20 2018 *)
%t A299822 f[p_, e_] := (p-1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 19 2020 *)
%o A299822 (PARI) a(n) = eulerphi(n)*factorback(factorint(n)[, 1]); \\ _Michel Marcus_, Jun 24 2019
%Y A299822 Cf. A000010, A007947, A065485, A173557, A323333.
%K A299822 nonn,mult,easy
%O A299822 1,2
%A A299822 _R. J. Mathar_, Feb 19 2018