A203639 Multiplicative with a(p^e) = e*p^(e-1).

1, 1, 1, 4, 1, 1, 1, 12, 6, 1, 1, 4, 1, 1, 1, 32, 1, 6, 1, 4, 1, 1, 1, 12, 10, 1, 27, 4, 1, 1, 1, 80, 1, 1, 1, 24, 1, 1, 1, 12, 1, 1, 1, 4, 6, 1, 1, 32, 14, 10, 1, 4, 1, 27, 1, 12, 1, 1, 1, 4, 1, 1, 6, 192, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 10, 4, 1, 1, 1, 32, 108, 1, 1, 4, 1, 1, 1, 12, 1, 6, 1, 4, 1, 1, 1, 80, 1, 14, 6, 40

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Krassimir Atanassov, New integer functions, related to φ and σ functions, Bull. Number Theory Related Topics, Vol. 11, No. 1 (1987), pp. 3-26.

G. L. Cohen, D. E. Iannucci, Derived Sequences, J. Int. Seq. 6 (2003) #03.1.1

Vaclav Kotesovec, Graph - the asymptotic ratio (500000000 terms).

J. Sandor, B. Crstici, Handbook of Number Theory II, Kluwer, 2004, page 337.

Formula

a(n) = n*A005361(n)/A007947(n).

a(n)=1 for all squarefree n.

Dirichlet g.f.: zeta^2(s-1)*product_{primes p} (1-2*p^(1-s)+p^(2-2s)+p^(-s)). - R. J. Mathar, Jan 19 2012

a(n) = A005361(n)*A003557(n). - Vaclav Kotesovec, Jun 20 2020

Maple

A203639 := proc(n)
    local a, f, e ;
    a :=1;
    for f in ifactors(n)[2] do
        e := op(2, f) ;
        p := op(1, f) ;
        a := a*e*p^(e-1) ;
    end do;
    a;
end proc; # R. J. Mathar, Jan 11 2012

Mathematica

Table[n*Times @@ Transpose[FactorInteger[n]][[2]] / Last[Select[Divisors[n], SquareFreeQ]], {n, 1, 100}] (* Vaclav Kotesovec, Dec 18 2019 *)
f[p_, e_] := e*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

Prog

(PARI) a(n)=my(f=factor(n)); n*prod(i=1, #f~, f[i, 2]/f[i, 1]) \\ Charles R Greathouse IV, Dec 09 2016
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X/(1 - p*X)^2))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020
(Scheme, with memoization-macro definec)
(definec (A203639 n) (if (= 1 n) n (* (A067029 n) (expt (A020639 n) (+ -1 (A067029 n))) (A203639 (A028234 n)))))
;; Antti Karttunen, Sep 13 2017

Crossrefs

Cf. A005361, A007947, A203640 (cycles).

Sequence in context: A063851 A124777 A352834 * A358319 A265679 A348989

Adjacent sequences: A203636 A203637 A203638 * A203640 A203641 A203642

Keyword

nonn,mult,easy

Author

R. J. Mathar, Jan 04 2012

Extensions

Terms a(1)-a(24) confirmed and terms a(25)-a(100) added by John W. Layman, Jan 04 2012

Status

approved