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

1, 2, 6, 4, 20, 12, 42, 8, 18, 40, 110, 24, 156, 84, 120, 16, 272, 36, 342, 80, 252, 220, 506, 48, 100, 312, 54, 168, 812, 240, 930, 32, 660, 544, 840, 72, 1332, 684, 936, 160, 1640, 504, 1806, 440, 360, 1012, 2162, 96, 294, 200, 1632, 624, 2756, 108, 2200, 336, 2052, 1624

Offset: 1

Author

R. J. Mathar, Feb 19 2018

Keywords

• nonn
• mult
• easy

Comments

A permutation of A323333. - Amiram Eldar, Sep 19 2020

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A007947, A065485, A173557, A323333.

Programs

Maple

A299822 := proc(n)
    local a,p,e,pe;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ; e:= pe[2] ;
        a := a*p*(p-1)*p^(e-1) ;
    end do:
    a ;
end proc:
seq(A299822(n),n=1..130) ;

Mathematica

Array[EulerPhi[#] SelectFirst[Reverse@ Divisors@ #, SquareFreeQ] &, 58] (* Michael De Vlieger, Feb 20 2018 *)
f[p_, e_] := (p-1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

PARI

a(n) = eulerphi(n)*factorback(factorint(n)[, 1]); \\ Michel Marcus, Jun 24 2019

Formula

a(n) = A000010(n)*A007947(n) = n*A173557(n).

Dirichlet g.f.: zeta(s-1)*Product_{p prime} (1 - 2*p^(1-s) + p^(2-s)), corrected by Vaclav Kotesovec, Dec 18 2019

Multiplicative with a(p^e) = p*(p-1)*p^(e-1).

a(n) = n*abs(A023900(n)). (Trivially rephrasing a formula in A173557.) - Omar E. Pol, Feb 19 2018

a(2^e) = 2^e. (Special case of above.) - Omar E. Pol, Feb 19 2018

A003557(n) | a(n). - R. J. Mathar, Feb 26 2018

From Vaclav Kotesovec, Dec 18 2019: (Start)

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)).

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)

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p-1)^2) = 2.826419... (A065485). - Amiram Eldar, Sep 19 2020

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