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%I #45 Mar 11 2023 10:39:42
%S 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,
%T 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,
%U 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
%N Multiplicative with a(p^e) = e*p^(e-1).
%H Antti Karttunen, <a href="/A203639/b203639.txt">Table of n, a(n) for n = 1..10000</a>
%H Krassimir Atanassov, <a href="https://catalog.hathitrust.org/Record/000681926">New integer functions, related to φ and σ functions</a>, Bull. Number Theory Related Topics, Vol. 11, No. 1 (1987), pp. 3-26.
%H G. L. Cohen, D. E. Iannucci, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Cohen/cohen6.html">Derived Sequences</a>, J. Int. Seq. 6 (2003) #03.1.1
%H Vaclav Kotesovec, <a href="/A203639/a203639.jpg">Graph - the asymptotic ratio (500000000 terms)</a>.
%H J. Sandor, B. Crstici, <a href="http://cdsweb.cern.ch/record/829127">Handbook of Number Theory II</a>, Kluwer, 2004, page 337.
%F a(n) = n*A005361(n)/A007947(n).
%F a(n)=1 for all squarefree n.
%F 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
%F a(n) = A005361(n)*A003557(n). - _Vaclav Kotesovec_, Jun 20 2020
%p A203639 := proc(n)
%p local a,f,e ;
%p a :=1;
%p for f in ifactors(n)[2] do
%p e := op(2,f) ;
%p p := op(1,f) ;
%p a := a*e*p^(e-1) ;
%p end do;
%p a;
%p end proc; # _R. J. Mathar_, Jan 11 2012
%t Table[n*Times @@ Transpose[FactorInteger[n]][[2]] / Last[Select[Divisors[n], SquareFreeQ]], {n, 1, 100}] (* _Vaclav Kotesovec_, Dec 18 2019 *)
%t f[p_, e_] := e*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 21 2020 *)
%o (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
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X/(1 - p*X)^2))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020
%o (Scheme, with memoization-macro definec)
%o (definec (A203639 n) (if (= 1 n) n (* (A067029 n) (expt (A020639 n) (+ -1 (A067029 n))) (A203639 (A028234 n)))))
%o ;; _Antti Karttunen_, Sep 13 2017
%Y Cf. A005361, A007947, A203640 (cycles).
%K nonn,mult,easy
%O 1,4
%A _R. J. Mathar_, Jan 04 2012
%E Terms a(1)-a(24) confirmed and terms a(25)-a(100) added by _John W. Layman_, Jan 04 2012
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